Although skate parks are intimidating enough at the best of times, and turning up on a mountain bike (faux pas #1) as a complete beginner (#2) and hogging the foam pit (#3) is probably not going to win you any friends.
You've correctly identified that big MTB-style jumps are a different kettle of fish, but they're not THAT different. If you're comfortable in the air that's the number 1 thing. The main difference is that proper trails kick you upwards more forcefully so you've got to be more fluid. As said, you need to ditch the SPDs, and getting a jump-specific hardtail is going to make things way easier than a bodged XC bike. The main issue I found (which is why I suck at dirt jumps) is that you need the balls to actually hit the jumps. I think because trails are typically slower, you can bottle it more easily than you can big jumps as part of a DH course, for example.
I also wouldn't say that being able to ride BMX-style trails makes you a 'better' rider - I've been riding with people who are ace at trails but can't really ride fast MTB tables. It's a different skill, it's not about one being better.