In my field (of physics) the formula $$ e^{-X} Y e^X = Y + [Y,X] + \frac{1}{2!} [[Y,X],X] + \ldots $$ for square matrices $X$ and $Y$ is erroneously called the "Baker--Campbell--Hausdorff (BCH) expansion", while it is really a lemma leading up to the "proper" BCH formula. In Brian C. Hall (2015), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, it is stated in Proposition 2.25 on page 48.
What is the proper name for the above expansion, if any?
I don't know that it has a name in mathematics; this is a formal consequence of the fact that the adjoint representation of $GL_n$ differentiates to the adjoint representation of $\mathfrak{gl}_n$, so the adjoint representation of the Lie algebra exponentiates to the adjoint representation of the Lie group; that is, $\exp(\text{ad}_X) = \text{Ad}_{\exp(X)}$.
If I had to give it a name I might call it "exp and ad commute" or "baby BCH."