A group is a set with a single operation defined on it that abides by certain axioms. A field is a set with two operations defined on it. But the field operations must abide by more axioms than a group operation. While vector spaces and fields are very similar (two operations, similar number of axioms), vector spaces are defined over fields (for example, an element cv is defined where v is a vector and c is a scalar) while fields are not defined over anything -- they are just a structure with two operations and a number of axioms (there's no element cv in a field, but it does have an element c_1 * c_2 where c_i is a scalar).
That said, vector field is a different object. It's a function. Likely named so by physics people while the structures like group/ring/field/etc were named by math folk.
I have no idea what flow maps of vector fields are, but if you give me their definition, it'd be trivial to check if they form a semigroup under a certain operation: we'll just check it for associativity.
To get a hang of this stuff I recommend the following books:
Book of Proof by Richard Hammack (tools of the trade)
Linear Algebra by Kuldeep Singh (rigorous tutorial: combines the rigor of a textbook and the ease of use of tutorial)
Abstract Algebra by the Dos Reis (rigorous tutorial)
Real Analysis by Lara Alcock (this books makes the rigorous definition of sequences trivial)
Real Analysis by Jay Cummings (contains much more info than the one above and is very similar in spirit)
Real Analysis by Rafi Grinberg (takes you from reals to Euclidean Spaces and Metric Spaces)
After that you ccan start reading intro level mathematical physics books to get an easy intro to differential geometry, manifolds and analysis in abstract spaces. Once you get an intuitive hang of this stuff, you can come back to the more brutal pure math setting.
Here, I like Modern Math Physics by Peter Szekeres. It's gentle and more about geometry and less about analysis.