I'm having trouble locating a single source that explains when the names of the books of the Torah occurred, as that would strengthen and shorten my answer. But, I'll provide some sources that I believe support my theory that the style of referring to the 5 books by their 1st name is a carry over from naming parshiot this way.
First, there's this article. It's very long, but on p. 71, it says:
In fact, nowhere in the Gemara is it even mentioned that the Torah is
partitioned into 54 parshiyos. The only requirement in the Gemara
concerning Shabbos Torah readings is limited to the need to read
certain portions of the Torah prior to Shevuos and Rosh Hashanah,
So, there's an implication that there was some other way to refer to the names of the 5 books, which I shall explain, later. Refer to footnotes 6 & 7 - they're important!
Further, on p. 71 going into p. 72, he cites Ramba"m תפילה יג׃ב where Ramba"m uses the opening words of a section of the Torah, a different pattern than what we call the parshiot, today.
Next, there is this source which shows the original names used for the 5 Books. Unfortunately, the author does not cite a source for these names. However, 2 of the names, Torat Kohanim and Mishnah Torah, IIRC, are mentioned in the Gemarah, and I assume that the others are, as well.
Lastly, keep in mind, that the Torah itself was written on a continuous scroll, so, obviously, there originally was no concept of "weekly" readings, initially. Nonetheless, from the way it is written, we certainly can determine that there were 5 "Books", though, the Torah itself, makes no mention of any divisions. I assume that this division was Masorah, then. (Worthy of a separate question on this, alone.) At any rate, this lends support to the notion that historically, there were different names for the 5 books of the Torah that DID refer to its content, similar to what is done for most of the Talmud. It seems that only years later, when a parsha system was developed, that the usage of the opening significant word was used for parshiot, and, thus it most likely carried over to the names for the 5 books, as well.