I would say that in games like Souky, gameplay is primarily 2D but with some 3D elements.
It's essentially a 2d game but with two Z layers (one level is the same as your ship, and the other one is the off-level) but only for some of the gameplay mechanics.
For instance, in Souky, enemies and your shot exists in two Z planes. If the enemies are at the same level as your ship, you can collide with them and hit them with any attack.
If the enemies are not on the same level you can only hit them with the lock-on weapon.
Dodging bullets, however, is entirely 2D (correct me if I'm wrong ofc). In other words, enemy bullets are only on the same level as your ship so there is no Z depth there.
To me that is much more closer to a pure 2D game than a pure 3d game. Xevious, Rayforce, and others I think are similar.
Space Harrier however, dodging occur in virtually 3D. Bullets have a near-continuous trajectory as they get closer and closer to you in terms of depth (Z) as well as moving along the X and Y axis. So here were are not talking about two 2D layers, but so many layers that it looks more like a Continuum--and therefore this game is closer to 3D than 2D gameplay. And both shooting and dodging is virtually 3D in Space Harrier.
Maybe there is an accepted scholar definition? Otherwise, a definition like this maybe would work?
A shmup is a game in which gameplay mechanics are primarily two-dimensional with focus on shooting and dodging, and in which movement is either one-dimensional or moving freely across two dimensions.
The other approach would be to take a family resemblance view and consider a number of elements to belong to a shmup and the more elements are there in a certain game, the more prototypical shmup it would be.
2D shooting mechanics
2D dodging mechanics
Top-down or side point of view
Flying character (helicopter, space ship, witch on a broom etc)
Fixed screen or Auto scrolling
Waves or Stages that can be cleared
Hi score function