Spider-Man: Homecoming is part of the Marvel Cinematic Universe, typically identified as Earth-199999. Like most of the other MCU characters, the MCU Spider-Man is a brand new character, that's an amalgamation of comic source material from various comic universes, plus original material created for the movies.
The two primary sources for material in the MCU are the mainstream "616" Marvel universe, and the Ultimate "1610" universe. The typical pattern for these movies seems to be that they pull very heavily from the Ultimates universe, both in character details and in tone.
In Spider-Man's case, it's hard to tell because so much of his character history is the same between the two universes. The key differences in Peter's backstory involve the identity of his parents and the specifics of where he got his powers, two things we have yet to see in the movies.
The Ultimates universe is overall has a more "modern" tone, which this Spider-Man movie certainly does as well. There are also specific hints of the Ultimates universe in there -- MJ being a schoolmate of Peter's, for example, and the hints that he might date MJ before Gwen (assuming Michelle is a stand-in for Mary Jane). On the other hand, Ultimate Peter Parker was never (I don't think) a member of the Avengers; also, Vulture is obviously using his mainstream identity of Adrian Toomes.
One major difference between he MCU and both comic universes is the total lack of the Fantastic Four and the X-Men. In the comics, Peter and Johnny Storm are very closely linked, and in the Ultimates universe, he's also classmates with Kitty Pryde. The fact that these characters can't appear in any Sony/Marvel movies means that they will have to change Peter's story going forward.
There will always be opportunities to call back to classic comic book moments, like the one you mentioned, but the MCU writers aren't going to try to "copy" Amazing Spider-Man or Ultimate Spider-Man on-screen; they'll take the ideas they like from the comics and use them to craft a brand new version.