Does all that really add up to being called a pro kit though?
It could be used as a pro kit for sure, but for me, the user determine if it's pro stuff or not.
There a difference between calling a given model a "pro instrument" and the status of its owner, by default a hobbyist who gig 4 times a year and play solely for pleasure can hardly call himself or his kit as professional, in contrast, the seasoned drummer who's touring and gigging 5 days a week can certainly tag himself and his gear as professional.
Is it the gear or the money spent?
In whichever case, it's both, isn't it?
Do you have to spend over 4k to make it professional or do only certain brands get the right to be called professional?
No, but so much depends how much you need to express yourself, a pro drummer with a basic 4 piece kit and 3 cymbals could probably get away with less than 4k per complete kit, and a hobbyist can spend much more for his 12 piece drumset, even if his kit is not a top of the line drums or cymbals, sizes do matters
How do you guys classify your gear and what is professional?
My drums and cymbals are referred as "pro instrument" or "top of the line model", and it can be used and has been used by many pro drummers, but I'm no pro drummer.
Manufacturers provides instruments, at different level of quality, craftsmanship and price tags. It's the buyers which makes the "professional" tag or reputation.
You can have a top of the line kit sounding like crap, because of its owner.
You can have a middle of the line kit sounding like a million dollar's kit, because of its owner.
It's all relative really, a pro drummer wants a kit he/she can tear down and set up night after night, and everything stays in perfect working orders.