The strict press will always refer to the shoulder press, as the strict part refers to there not being any work done by the legs or torso, and only the deltoids and triceps power the movement.
Because the chest press is performed laying down, no momentum can be generated with the rest of the body, hence, strict is only applicable to the shoulder press.
We can assume that when a reference to a press is made and not a STRICT press, it can mean that the press can be performed in any matter, i.e. just bring the weight from a racking position to overhead.
At Cavemantraining we name a press that is powered by muscles that move the thoracic spine a spinal press (not to be confused with the spiral press). It might sound weird, but it makes the most sense. When the athlete bends back (thoracic spine hyperextension) then that range in the shoulder is created by the spine movement. The athlete then returns the spine to neutral/straight, and the muscles that moved the spine also moved the weight. More about this in the Spinal press.
The press can be powered by any muscle that moves joints in the legs and spine, but when one does so, it's no longer a strict press.