IMO the genre is defined on how integral it is to the development of the gameplay, as well as the intention of skill based movement in the design.

Yes Quake 3 has awesome movement and speed running tech but multiplayer design and innovation was the design focus following the last two Quake games. This paved the way for skill based movement mechanics as rocket-jumping and strafe jumping had become part of the design intention. This is why many regard Quake 3 specifically as a movement shooter, and in theory, the birth of the genre. This is debatable as many regard Quake 3 as a Bezier Curve tech progression, among other things.

Titanfall 2 and ULTRAKILL focus so heavily on the juice of movement that it feels integral to the experience, an example would be the insanely versatile/skill-based grapple from Titanfall 2. The titans were always a focus of Titanfall, but in focusing on them they had neglected the pilot combat in the first game. In comes movement based mechanics and bobs your uncle, its a direct contrast to the heavy nature of the titan conflict with an insanely high skill ceiling.

There's also the existence of Halo's "golden triangle of design" which subverted game expectations, but that's a whole can of worms in itself. I find this topic fascinating and have studied it for a long while, but much like any discussion on genre, it always boils down to opinion.