Steinberg's "Lectures on Chevalley Groups" 
<https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf>
contain ``a complete list of isomorphisms" among the various finite simple Chevalley groups (Th. 37 on pp. 108--109). Unfortunately, the proofs are omitted. I am looking for a reference where the completeness of the list is proven.



Exceptional isomorphisms between finite simple Chevalley groups
I've tracked down, I think, the best references although I don't have access to them. A description of the history of this question is in Wilhelm Magnus' preface to the Dover edition of Dickson's *Linear groups*:



> 
> In a later paper Dieudonné settled one of the fundamental questions
>  which Dickson had left unanswered by showing that Dickson's list of
>  isomorphisms between the simple groups discussed in his book is
>  complete. Finally, in 1955, E. Artin, in two astonishingly short
>  papers, demonstrated that the whole table for the orders of finite
>  simple groups and the isomorphisms between them can be derived
>  systematically, with discussion of only a very few separate cases.
> 
> 
> 


The references mentioned by Magnus are below.


*Jean Dieudonné*, MR 45125 [**On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua**](http://www.ams.org/mathscinet-getitem?mr=45125), *Mem. Amer. Math. Soc.,* **1951** (1951), no. 2, vi+122.


*Emil Artin*, MR 70642 [**The orders of the linear groups**](http://www.ams.org/mathscinet-getitem?mr=70642), *Comm. Pure Appl. Math.* **8** (1955), 355--365.


*Emil Artin*, MR 73601 [**The orders of the classical simple groups**](http://www.ams.org/mathscinet-getitem?mr=73601), *Comm. Pure Appl. Math.* **8** (1955), 455--472.


Note that these references predate the discovery of some of the finite simple groups (e.g. Suzuki's) so they won't deal with all of the finite groups of Lie type. But, given you only want the Chevalley groups, you should be fine.


