The term Chevalley fundamental group is sometimes used for the fundamental group of a space defined via the automorphisms of a universal cover.
This terminology is used by Borceux and Janeldize in their book on Galois Theories. To quote from that source:
” There are two classical definitions of the fundamental group of a topological space which give isomorphic groups for certain ”good” spaces.”
The Chevalley fundamental group $\mathrm{Aut}(p)=\mathrm{Aut}(E,p)$ is defined only for connected spaces $B$ which admit a universal covering map $p:E\to B$ with connected $E$, and, of course, depends on it, but again, different $p$ produce isomorphic groups.”
The term algebraic fundamental group is also sometimes used for this, although more usually that term is reserved for Grothendieck’s fundamental group of a scheme.
Francis Borceux, George Janelidze, Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.
C. Chevalley, Theory of Lie groups, Princeton University Press, 1946.