Primary lift is the focus of that specific training cycle. Since I powerlift, it can be competition squat, bench, or deadlift or it can be something else but very similar to the competition lifts. If I compete raw no wraps then my primary lift can be a raw squat no wraps or if I want something different it can be squats with wraps. My competition bench is paused so a primary bench can be a paused bench or maybe something like a touch and go bench. For a bodybuilder or anyone else a primary lift can be any lift they want to get stronger on or a lift to focus on a weak area.
Secondary lift is any lift that's not your primary lift for that cycle.
Competition lift means the lifts I'll do in a powerlifting meet, exactly how I'll do them in a powerlifting meet. If I squat without wraps but with a belt and wrist wraps then my competition squat is a squat without wraps but with a belt and wrist wraps. If I do conventional deads then by competition deadlifts will be conventional deads and not sumo for example. You may not have access to the same bars they have at meets so if you use a standard Olympic bar in the gym but compete with a DL bar then keep the Olympic bar bc that's all you have.
Assistance lifts are anything to help drive your main or primary lifts. They typically are worked with the same/similar intensities and reps but maybe we change the ROM. A deficit dead, rack pull, board press, floor press, box squat, etc can all be assistance lifts.
Supplement lifts are further away in similarity from your primary lifts. Intensities are usually lower, reps are higher, volume can be added to them, they can be lifts to work the same muscles as the primary lift but in a different plain of motion. Examples could be front squats, incline bench, dips, snatch grip deadlifts, etc.
A particular lift can be used for more than one slot it just depends on how you want to setup your training, what your goals are in general, and what your focus is for the cycle.
