I am reasonably sure you have quoted everything relevant that D&D 5e has to offer.
However, based on hints in 5e, and in some of the precedents from older D&D content, my understanding of the arcane trickster (and the eldritch knight) is that the spells come from intense study, similar to a wizard, but highly specialized—rather than learning the entire system of magic and the ability to read a spellbook and prepare different spells every day, these characters instead memorize a very specific few spells to go alongside their other focuses outside of magic.
As you note, I think 5e is already hinting at this just from the fact that these classes use Intelligence for their magic, and what Intelligence is defined as.
However, fifth edition was not the debut of the arcane-magic-casting rogue known as the “arcane trickster.” In D&D 3.5e, the arcane trickster was a “prestige class,” a class you couldn’t start out in, but had to multiclass into after meeting its requirements. This edition didn’t have “subclasses” in the same way that 5e does, and prestige classes were the much more common way to add a little bit of another class to your character. In the arcane trickster’s case, it allowed a rogue/wizard or rogue/sorcerer to progress both their sneak attack and their spellcasting at the same time.
Since you had to already have spellcasting to become an arcane trickster, there was never any question of where that spellcasting came from—if you were a rogue/wizard/arcane trickster, you studied wizardry; if you were a rogue/sorcerer/arcane trickster, you had sorcery in your blood. If you were a very strange bard/rogue/arcane trickster—or even weirder, an assassin/arcane trickster, a combination I only mention because the 3.5e Dungeon Master’s Guide does—then your arcane spellcasting was just as it would be for a bard, or for an assassin, because you were a bard or assassin.
Thus, since the typical Intelligence-based arcane trickster was a rogue/wizard, it implies that the Intelligence-based spells of the 5e arcane trickster is also wizardry. In fact, since there were only so many classes that could qualify for arcane trickster and provide Intelligence-based arcane spellcasting, and almost all of them were spellbook-using prepared spellcasters very much like the wizard, this implication was quite strong.
Of course, the 5e arcane trickster is not (necessarily) a rogue/wizard, and even if they are their arcane trickster spells and wizard spells are separate. And arcane tricksters use their spells differently from wizards, since they learn only a select few spells rather than have the ability to collect a whole spellbook and prepare exactly what they need each day. D&D 3.5e did have spellcasters like that—the aforementioned assassin, the eldritch knight-like duskblade, and most relevantly, the beguiler, who combined rogue-like sneaking and social skills with a powerful mastery of enchantment and illusion. Unfortunately, none of these really delves deeply into how their spells really work. The closest we get is the tiniest tidbit for the beguiler, who had a special class feature to learn a few extra spells, “representing the result of personal study and experimentation.”
