Are these three representations for compact Lie groups and its Lie algebra mean the same thing? Let's focus on "classical" cases like $SO(N)$ and $\mathfrak{so}(n)$ for concreteness (unless I need to go beyond these to make the distinction clear).
Based on my reading of several physics-oriented texts, at the very least defining representation and regular representation mean the same thing (e.g. Georgi's text on Lie algebra for particle physics). That is, they are the representations given by "identity" map, since these are matrix groups/algebras. On the other hand, if I pick a book like Gauge/Gravity duality by Erdmenger, it seems that fundamental representation for $SO(N)$ is the one with dimension $N$, which would then coincide with the defining representation (and hence regular representation). However, I believe by definition, fundamental representations are not just defined by its dimensionality (more like in terms of highest weights and all that), so it should be different in general though coincidences may occur. I think for $N=3$, the defining representation (as I understand above) even coincides with adjoint representation (but at least in general adjoint representation has very different dimensions).
As a side question, do terminologies from Lie groups and Lie algebras agree? I mean, sure, the matrices may be very different ($GL(n)$ being unit determinant but its Lie algebra being any square matrix), but the "dimensionality" I believe should agree (unless I am wrong, which is why I am asking).
I know (roughly) the formal definition of fundamental representation, but much less so on the other two and hence the confusion of the use of terms in mathematics. I would appreciate either (1) formal definitions, (2) clear distinctions between them, (3) explicit (counter)examples of representations which demonstrate their differences, and/or (3) resources which clearly define these three.
