I propose that this is what you "really" mean to say by 2): "what is the maximum length $t(L)$ such that for any $0$-dimensional scheme $Z$ of length $t(L)$, the evaluation morphism $H^0(L) \otimes \mathcal{O}_X \to L_{|{_Z}}$ is surjective". This, I think, is what you mean by letting the points come together. For an general very ample line bundle $L$, the answer is of course $t(L)=2$ (very ampleness is equivalent to $t(L) \geq 2$, and it is not hard to find examples where $t(L)=2$).

Here is the exact terminology I think you are looking for: a line bundle is called $k$ very ample if $t(L) \geq k+1$. This is a notion due to Sommese and there is a huge amount of literature about it. A good primer on it is Ch. 5 of Göttsche's paper "A conjectural generating function for numbers of curves on surfaces". This is closely related to the notion of Seshadri constants as in Karl Schwede's answer, but it is slightly different in general.

For example a globally generated line bundle is $0$ very ample and a very ample line bundle is $1$ very ample.

Seeing as you were taking powers of $L$, perhaps the result you are looking for is the following: suppose $L$ is $k$ very ample, then $L^m$ is $km$ very ample. In particular, if $L$ is very ample, then the answer to 2) in general, with my interpretation of the question, is $t_0=m+1$.

I propose that this is what you "really" mean to say by 2): "what is the maximum length $t(L)$ such that for any $0$-dimensional scheme $Z$ of length $t(L)$, the evaluation morphism $H^0(L) \otimes \mathcal{O}_X \to L_{|{_Z}}$ is surjective". This, I think, is what you mean by letting the points come together. For an general very ample line bundle $L$, the answer is of course $t(L)=2$ (very ampleness is equivalent to $t(L) \geq 2$, and it is not hard to find examples where $t(L)=2$).

Here is the exact terminology I think you are looking for: a line bundle is called $k$ very ample if $t(L) \geq k+1$. This is a notion due to Sommese and there is a huge amount of literature about it. A good primer on it is Ch. 5 of Göttsche's paper "A conjectural generating function for numbers of curves on surfaces". This is closely related to the notion of Seshadri constants as in Karl Schwede's answer, but it is slightly different in general.

For example a globally generated line bundle is $0$ very ample and a very ample line bundle is $1$ very ample.

Seeing as you were taking powers of $L$, perhaps the result you are looking for is the following: suppose $L$ is very ample, then $L^m$ is $m$ -very ample. In particular, if $L$ is very ample, then the answer to 2) in general, with my interpretation of the question, is $t_0=m+1$.

EDIT: I originally stated that $L$ $k$-very ample implies $L^m$ is $km$ very ample. This does seem to be true but may be less well-known than I recalled: it follows from the paper of Hinohara, Takahashi, Terakawa "On tensor products of $k$-very ample line bundles". For line bundles on K3 surfaces (where I usually work) or Enriques surfaces, the result is more well-known; it follows from Knutsen's paper "On kth order embeddings.." plus Lemma 0.5.3 of Beltrametti and Sommese, "On k-spannedness for projective surfaces". The weaker statement $L$ very ample implies $L^m$ $m$-very ample, which is all we need, follows from Beltrametti and Sommese, "On k-jet ampleness".

It is worth having a quick look at these papers above. It turn out there are three related notions related to how one lets points come together: $k$ very ample, $k$ spanned and $k$ jet ample, which can sometimes coincide but not always. In general, $k$ jet ample is the strongest, $k$ spanned is the weakest and $k$ very ample sits in the middle. For $k=1$ they all coincide. For some surfaces like K3s, $k$ spanned and $k$ very ample are the same.