The list of finite *simple* groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from different Lie types). Call this collection of isomorphism classes of finite groups $\mathcal{S}$. But it's not clear to me that there is a similar consensus about the meaning of "finite group of Lie type". Probably most people would place the finite general linear groups on this list. Maybe also the Weyl group of the $E_8$ root system, which is related in the *Atlas of Finite Groups* to the simple group denoted $G=\mathrm{O}_8^+(2)$ as the group $2.G.2$.

The term "finite group of Lie type" comes up frequently in the literature (almost 500 times in a MathSciNet search of titles and reviews). There are two main directions in which such groups are approached, which might possibly lead to different lists. Often the starting point is a simple algebraic group over a finite field (with no nontrivial proper closed connected normal subgroups), though $\mathrm{GL}_n$ doesn't quite fit here.

(1) Steinberg's efficient organization in terms of finite fixed point groups under endomorphisms, denoted $G_\sigma$, is now often expressed in terms of "Frobenius morphisms" and their fixed points $G^F$. Basically the groups of interest then come in three flavors: split (Chevalley) groups, quasi-split (Hertzig, Steinberg, Tits) groups in types $A_n (n \geq 2), D_n, E_6$, or Suzuki/Ree groups in types $B_2, G_2, F_4$ with $p = 2, 3, 2$ as defining characteristic.

This leads to a collection $\mathcal{L}_1 \supset \mathcal{S}$ of (isomorphism classes of) Lie-type groups if one is allowed to take derived groups and to factor groups by subgroups of their centers. For this I certainly want to include general linear and unitary groups, so for type $A_{n-1}$ the collection $\mathcal{L}_1$ includes $\mathrm{GL}_n, \mathrm{SL}_n, \mathrm{PGL}_n, \mathrm{PSL}_n$ and related unitary groups. Often people specify at the outset a connected *reductive* group $G$ over a finite field, but it's best to assume that $G$ has a simple derived group for the current purpose.

(2) The *Atlas* focuses instead on the finite simple groups and then builds character tables for various related groups, as in my Weyl group example above. Here the simple group $G$ can be enlarged to a group in which $G$ is normal and the quotient is abelian, or can acquire nontrivial central extensions using its Schur multiplier. Starting with the collection $\mathcal{S}$, one obtains in this way a larger collection $\mathcal{L}_2$ of (isomorphism classes of) "groups of Lie type". Partly because of questions about general linear groups and the like, I'm left with a question:

How close to being the same are the collections $\mathcal{L}_1$ and $\mathcal{L}_2$? (Is there a natural way to define them to ensure equality?)