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 is defined only for connected spaces which admit a universal covering map with connected , and, of course, depends on it, but again, different 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.