Can anyone help me derive and/or define the infinitesimal generators of a Lie group?

Question

I'm doing a project involving Lie groups in Physics, and a part of the project involves generators. I initially used Robert Gilmore's 'Lie Groups, Lie Algebras, and some of their Applications'which gives a very detailed account ending with the definition $$ X_\mu (x') =-\frac{\partial}{\partial q^\mu}(f^j (q^\mu, x'(p)))\bigg|_{q=0} \frac{\partial}{\partial x'^\mu}$$ where X is the infinitesimal generator of a Lie group, with q being some dummy variable, with X going in $$\delta F = \delta^\mu X_\mu (x') F^S(x'(p)),$$ where $F$ is a transformation in the geometric space. However, according to my thesis supervisor, my explanation inspired by this is not completely correct (for example, it treats a Lie group as its own geometric space, even though the geometric space is a vector space, whereas Lie groups cannot be assumed to be vector spaces). I also tried to instead explain things in terms of Killing vectors, but he noted that Killing vectors are representations of generators, rather than generators themselves, and therefore that this is not entirely correct. Can anyone help me give a better account of what exactly the infinitesimal generators of Lie groups are, and perhaps also direct me to a text that explains this more clearly than Robert Gilmore's textbook?