I can't answer your general question but I can answer your side question. Almost all of the groups of Lie type are constructed as follows. You take a simple algebraic group $G$ defined over an algebraic closure of the prime field $\overline{\mathbb{F}_p}$ for some prime $p>0$. You then take a generalised Frobenius endomorphism $F : G \to G$ and consider the fixed point subgroup $G^F = \{g \in \mid F(g) = g\}$. Up to a very small number of exceptions the quotient $G^F/Z(G^F)$ is then a finite simple group, known as a finite simple group of Lie type. Tits's very elegant theory of BN-pairs allows you to show relatively easily that $G^F/Z(G^F)$ is simple. I actually think the order formula for the group as a polynomial in $q$ also allows you to deduce quick quickly that you mostly get a pairwise non-isomorphic list.

Now the simple simply connected algebraic groups are labelled using Lie theoretic notation. For instance, if $G$ is $\mathrm{SL}_n(\overline{\mathbb{F}_p})$ then this would be of Lie type $\mathrm{A}_n$. Now the generalised Frobenius endomorphisms on $G$ are essentially classified by pairs $(q,\phi)$ consisting of $q = p^a$ a power of $p$ and an automorphism $\phi$ of the Dynkin diagram of $G$. For instance, in type $\mathrm{A}_n$ there is exactly one non-trivial automorphism of the Dynkin diagram say $\tau$. One then has two sets of triples $(\mathrm{A}_n,q,\mathrm{id})$ and $(\mathrm{A}_n,q,\tau)$. The first corresponds to the infinite series $\mathrm{PSL}_{n+1}(q)$ consisting of the projective special linear groups and the second corresponds to the series consisting of the projective special unitary groups $\mathrm{PSU}_{n+1}(q)$. Obviously there are some known cases where these are not simple. For instance $\mathrm{PSL}_2(2) \cong \mathfrak{S}_3$ and $\mathrm{PSL}_2(3) \cong \mathfrak{A}_4$ are not simple but if I remember rightly these are the only examples in the family $\mathrm{PSL}_{n+1}(q)$.

Annoyingly there is one group of Lie type which cannot be described in this way, namely the Tits group. This is the derived subgroup of the Ree group ${}^2\mathrm{F}_4(2)$ and is usually denoted ${}^2\mathrm{F}_4(2)'$. Many people working in groups of Lie type would consider this to be a Sporadic group because of this reason. However, as everyone knows by now that there are 26 sporadic finite simple groups you would like quite stupid claiming there to be 27.

Edit: You may find the recent book "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman enlightening. Section 24 of the book specifically talks about constructing finite simple groups of Lie type in quite a readable way. Specifically have a look at Remark 24.9, Theorem 24.17 and Remark 24.18.

I can't answer your general question but I can answer your side question. Almost all of the groups of Lie type are constructed as follows. You take a simple algebraic group $G$ defined over an algebraic closure of the prime field $\overline{\mathbb{F}_p}$ for some prime $p>0$. You then take a generalised Frobenius endomorphism $F : G \to G$ and consider the fixed point subgroup $G^F = \{g \in G\mid F(g) = g\}$. Up to a very small number of exceptions the quotient $G^F/Z(G^F)$ is then a finite simple group, known as a finite simple group of Lie type. Tits's very elegant theory of BN-pairs allows you to show relatively easily that $G^F/Z(G^F)$ is simple. I actually think the order formula for the group as a polynomial in $q$ also allows you to deduce quick quickly that you mostly get a pairwise non-isomorphic list.

Now the simple simply connected algebraic groups are labelled using Lie theoretic notation. For instance, if $G$ is $\mathrm{SL}_n(\overline{\mathbb{F}_p})$ then this would be of Lie type $\mathrm{A}_n$. Now the generalised Frobenius endomorphisms on $G$ are essentially classified by pairs $(q,\phi)$ consisting of $q = p^a$ a power of $p$ and an automorphism $\phi$ of the Dynkin diagram of $G$. For instance, in type $\mathrm{A}_n$ there is exactly one non-trivial automorphism of the Dynkin diagram say $\tau$. One then has two sets of triples $(\mathrm{A}_n,q,\mathrm{id})$ and $(\mathrm{A}_n,q,\tau)$. The first corresponds to the infinite series $\mathrm{PSL}_{n+1}(q)$ consisting of the projective special linear groups and the second corresponds to the series consisting of the projective special unitary groups $\mathrm{PSU}_{n+1}(q)$. Obviously there are some known cases where these are not simple. For instance $\mathrm{PSL}_2(2) \cong \mathfrak{S}_3$ and $\mathrm{PSL}_2(3) \cong \mathfrak{A}_4$ are not simple but if I remember rightly these are the only examples in the family $\mathrm{PSL}_{n+1}(q)$.

Annoyingly there is one group of Lie type which cannot be described in this way, namely the Tits group. This is the derived subgroup of the Ree group ${}^2\mathrm{F}_4(2)$ and is usually denoted ${}^2\mathrm{F}_4(2)'$. Many people working in groups of Lie type would consider this to be a Sporadic group because of this reason. However, as everyone knows by now that there are 26 sporadic finite simple groups you would like quite stupid claiming there to be 27.

Edit: You may find the recent book "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman enlightening. Section 24 of the book specifically talks about constructing finite simple groups of Lie type in quite a readable way. Specifically have a look at Remark 24.9, Theorem 24.17 and Remark 24.18.

I can't answer your general question but I can answer your side question. Almost all of the groups of Lie type are constructed as follows. You take a simple algebraic group $G$ defined over an algebraic closure of the prime field $\overline{\mathbb{F}_p}$ for some prime $p>0$. You then take a generalised Frobenius endomorphism $F : G \to G$ and consider the fixed point subgroup $G^F = \{g \in \mid F(g) = g\}$. Up to a very small number of exceptions the quotient $G^F/Z(G^F)$ is then a finite simple group, known as a finite simple group of Lie type. Tits's very elegant theory of BN-pairs allows you to show relatively easily that $G^F/Z(G^F)$ is simple. I actually think the order formula for the group as a polynomial in $q$ also allows you to deduce quick quickly that you mostly get a pairwise non-isomorphic list.

Now the simple simply connected algebraic groups are labelled using Lie theoretic notation. For instance, if $G$ is $\mathrm{SL}_n(\overline{\mathbb{F}_p})$ then this would be of Lie type $\mathrm{A}_n$. Now the generalised Frobenius endomorphisms on $G$ are essentially classified by pairs $(q,\phi)$ consisting of $q = p^a$ a power of $p$ and an automorphism $\phi$ of the Dynkin diagram of $G$. For instance, in type $\mathrm{A}_n$ there is exactly one non-trivial automorphism of the Dynkin diagram say $\tau$. One then has two sets of triples $(\mathrm{A}_n,q,\mathrm{id})$ and $(\mathrm{A}_n,q,\tau)$. The first corresponds to the infinite series $\mathrm{PSL}_{n+1}(q)$ consisting of the projective special linear groups and the second corresponds to the series consisting of the projective special unitary groups $\mathrm{PSU}_{n+1}(q)$. Obviously there are some known cases where these are not simple. For instance $\mathrm{PSL}_2(2) \cong \mathfrak{S}_3$ and $\mathrm{PSL}_2(3) \cong \mathfrak{A}_4$ are not simple but if I remember rightly these are the only examples in the family $\mathrm{PSL}_{n+1}(q)$.

Annoyingly there is one group of Lie type which cannot be described in this way, namely the Tits group. This is the derived subgroup of the Ree group ${}^2\mathrm{F}_4(2)$ and is usually denoted ${}^2\mathrm{F}_4(2)'$. Many people working in groups of Lie type would consider this to be a Sporadic group because of this reason. However, as everyone knows by now that there are 26 sporadic finite simple groups you would like quite stupid claiming there to be 27.

I can't answer your general question but I can answer your side question. Almost all of the groups of Lie type are constructed as follows. You take a simple algebraic group $G$ defined over an algebraic closure of the prime field $\overline{\mathbb{F}_p}$ for some prime $p>0$. You then take a generalised Frobenius endomorphism $F : G \to G$ and consider the fixed point subgroup $G^F = \{g \in \mid F(g) = g\}$. Up to a very small number of exceptions the quotient $G^F/Z(G^F)$ is then a finite simple group, known as a finite simple group of Lie type. Tits's very elegant theory of BN-pairs allows you to show relatively easily that $G^F/Z(G^F)$ is simple. I actually think the order formula for the group as a polynomial in $q$ also allows you to deduce quick quickly that you mostly get a pairwise non-isomorphic list.

Now the simple simply connected algebraic groups are labelled using Lie theoretic notation. For instance, if $G$ is $\mathrm{SL}_n(\overline{\mathbb{F}_p})$ then this would be of Lie type $\mathrm{A}_n$. Now the generalised Frobenius endomorphisms on $G$ are essentially classified by pairs $(q,\phi)$ consisting of $q = p^a$ a power of $p$ and an automorphism $\phi$ of the Dynkin diagram of $G$. For instance, in type $\mathrm{A}_n$ there is exactly one non-trivial automorphism of the Dynkin diagram say $\tau$. One then has two sets of triples $(\mathrm{A}_n,q,\mathrm{id})$ and $(\mathrm{A}_n,q,\tau)$. The first corresponds to the infinite series $\mathrm{PSL}_{n+1}(q)$ consisting of the projective special linear groups and the second corresponds to the series consisting of the projective special unitary groups $\mathrm{PSU}_{n+1}(q)$. Obviously there are some known cases where these are not simple. For instance $\mathrm{PSL}_2(2) \cong \mathfrak{S}_3$ and $\mathrm{PSL}_2(3) \cong \mathfrak{A}_4$ are not simple but if I remember rightly these are the only examples in the family $\mathrm{PSL}_{n+1}(q)$.

Annoyingly there is one group of Lie type which cannot be described in this way, namely the Tits group. This is the derived subgroup of the Ree group ${}^2\mathrm{F}_4(2)$ and is usually denoted ${}^2\mathrm{F}_4(2)'$. Many people working in groups of Lie type would consider this to be a Sporadic group because of this reason. However, as everyone knows by now that there are 26 sporadic finite simple groups you would like quite stupid claiming there to be 27.

Edit: You may find the recent book "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman enlightening. Section 24 of the book specifically talks about constructing finite simple groups of Lie type in quite a readable way. Specifically have a look at Remark 24.9, Theorem 24.17 and Remark 24.18.