What whip on a lazy horse?

Have you made sure something isn't wrong or hurting your horse first like its teeth, the saddle or your position? A whip is only a extension of your arm and shouldn't be used to beat a horse. I think your instructor should give you some lessons on how your position could be impacting on the horse and the correct leg and arm aids you need to use. If anything use a dressage whip as a extension of your arm. I once rode a horse and the woman said he was "lazy" I got on him with no whip and he had the most amazing movement once the right aids were given to him.

Personally I pick depending on the horse.

I quite like a long whip since you don't have to let go of the reins. Which means you don't accidently drop your contact or pull back. It can also allow you to tap directly behind your leg (or with experience tickle the leg to encourage the horse to move that leg).

A short whip is easier to change over, easier to jump and hack with but you have to take your hand off the rein. And for a larger horse, it can just be more of a hassle.

I have both - my long whip stands out waist heigh on me (I'm 5ft 5). But it works wonders on one of the larger horses I ride. I have used it on one of the cob ponies too because she can be very unresponsive but has a sensitive mouth and has steering issues - so in balance, it's easier to keep hands on the reins. But for another horse who can be lazy and slow but also can bomb off and spook at nothing, I carry a short whip so I don't accidently waft it into his view and because tickle is usually enough in a lazy mood and I certaintly wouldn't want to use it at all when he is going.

Don't be afraid to try both - one may work on one horse better than on another.

Let your instructor guide you. Personally, I don't like crops at all, I like dressage whips, probably what you are calling a schooling whip. A longer whip lets you keep your hand in position, and still use the whip where you should, that is, behind your leg as reinforcement for your leg.

I prefer a shorter whip. I find my self accidentally tapping the horse with the long flexibe dressage whip.

I'd be more inclined to make sure this 'being lazy', isn't to do with something medical. Incorrect saddle ? etc. The whip is only meant to be a control aid.