My understanding is 2-descent today means calculating $E(\mathbb Q)/2E(\mathbb Q)$ by computing the Selmer group and trying to figure out which curves of the Selmer group actually have a K-point. (See Silverman's Arithmetic of Elliptic curves Chapter X Prop 1.4 or Prop 4.9)
I have heard that the name arose from Fermat's original proof by descent that there are no non-trivial (i.e. completely non-zero) solutions to the diophantine equation $x^4-y^4=z^2$. See here. As shown in the link, this equation can be interpreted as the elliptic curve $E$ given by $y^2 = x^3 + 4x$, the "trivial" points becoming the points $X = \{\mathcal O, (0,0), (2,4), (2,-4)\}$.
Then Fermat is proving there are no points besides $X$ on $E(\mathbb Q)$. Specifically, he proves any $P \in E(\mathbb Q) \backslash X$ satisfies $P = [2]Q$ for some $Q \in E(\mathbb Q)\backslash X$ and that the height of Q will be strictly smaller than the height of P. An elliptic curve has only finitely many points of bounded height, and so repeating this process leads to a contradiction.
I had thought the generation of smaller and smaller points was the "descent" part of Fermat's theorem. However, modern day 2-descent doesn't seem to be a generalization of that part of the proof? It doesn't involve a height, or shrinking points. It seems more analogous to the first step of the proof where Fermat figured out what X was. In the modern picture, descent seems to better describe the second step of the Mordell-Weil theorem where representatives of $E(\mathbb Q)/2 E(\mathbb Q)$ and the height function are used to find generators of $E(\mathbb Q)$.
Is this a bit of a misnomer/evolution of the word, or is there some other reason why computing $E/2E$ is being called descent?