Yup, just jump into the DC. No reason to play vanilla. The game rewards stealthing, non-lethal takedowns and thorough exploration (with goodies and alternate entrances) a lot more, but you can also run-and-gun if you're not feeling patient.
Deus Ex is a classic. If you end up liking HR, definitely go back and play Deus Ex. It's still up there in terms of player choice. Anything you decide to do at pivotal moments will be acknowledged by the game, without over-promising with a branching narrative only text based games can actually deliver.
Invisible War was massively overshadowed by it's predecessor. It's not trash, and it does some neat things with the narrative, but it's also not essential in any way.