At the time this change was known as AJAX, and GMail is listed as a pioneer: https://en.wikipedia.org/wiki/Ajax_(programming)
Squirrelmail, by contrast, rendered a new page for most user actions. Especially the viewing of messages.
As I recall there was a fight about getting it in and the creators called it XMLHttpRequest because XML was the hotness at the time and that got it past the project managers and PHBs.
-Webmail was already established as a "thing" when Gmail came out, even in Universities.
-Gmail wasn't particularly special from a UI perspective and was comparable to other offerings at the time.
-There was plenty of other web apps already that weren't Google products. In fact, I, personally, was writing one at the time GMail came in out.
Pre-AJAX, web-based services were just a series of forms, and the interactivity patterns harkened back to "smart terminal" [1] form-based use for mainframes in the 1970s. I believe the user you were responding to uses "application" to mean something approximating the desktop application experience of the 80s or the current mobile app experience.
So the distinction they're drawing is about the kind of interactivity. Squirrelmail, et al, were a series of forms and pages. GMail didn't do a new page or frame load every time you looked at a new message. UI rendering and interactivity became client-only activities, with the server providing an API.
[1] now known as block-oriented terminal: https://en.wikipedia.org/wiki/Block-oriented_terminal
GMail was different because it offered an outlandish 1GB of mail space, when everywhere else you got 1-10MB mailboxes. Since it was launched on April 1st, most people did not believe it at first. And (at the time) huge space allowed users to keep and search all old email, instead of deleting old emails being a weekly chore.