Well, just to be a pedantic sod, but you can indeed have a ski jump and a catapult on the same carrier, just not both at the same position.
It would certainly be possible, even desireable for the PLAN to have waist catapults and a bow ski jump on the same carrier.
China has already invested in 20+ J15As, and will probably get many more to form the Liaoning's air wing. That is no small investment.
They are also working on a catapult capable J15B.
A waist cat mod could very easily give the Liaoning the full benefit of cat assisted launches for strike loads and especially fixed wing carrier AWACS, while retaining the ski jump allows them to continue to use the non-cat capable J15As.
Once all the J15As are retired, they could do a major mid life upgrade to remove the ski jump and put in bow cats, but that's at least 15-20 years down the line.
Just regarding nomenclature, I think the non catapult compatible J-15s are just called "J-15s" whereas the upgraded catapult compatible J-15s are currently being called J-15A (or J-15T).
In either case, I wouldn't be surprised if production of vanilla J-15s ends within a year or so, replaced in production solely by the the upgraded and catapult compatible J-15A being standard (likely with J-15AS as well), to be the common aircraft aboard both STOBAR and CATOBAR carriers, while vanilla J-15s gradually become relegated to second line duties as more J-15As enter service (though this will be quite a few years away).
So I imagine J-15As will eventually operate from both STOBAR and CATOBAR carriers (very important for operational flexibility IMO) and it also streamlines production to a single standard -- there is no necessary obligation for them to keep operating a STOBAR carrier or anything to keep their current vanilla J-15s relevant, because any catapult J-15A should be able to operate from a STOBAR carrier just as competently.
Refitting Liaoning (and 001A, which is always part of the conversation) with waist cats is always going to be a persistent possibility, but I personally think the complexity and cost of such a modification may not be worth it, especially in Liaoning's case if the ship wasn't designed with a waist cat refit in mind. If 001A was designed and built with a waist cat refit in mind potentially, with the underdeck spaces and configuration arranged with minimal internal reconstruction needed then a waist cat refit might be more possible.