My theory is that classical piano will prepare you to branch off to anything later, whereas something specific like jazz will limit you to that style of playing. I know lots of primarily classical pianists who can play jazz, but hardly any primarily jazz pianists who can play classical.
This is so true! From my experience it's valid for other instrumentalists as well. The classical showed more interest in jazz than vice versa.
It is my opinion that as a musician you should learn music (reallly?!). You should go for everything that improves you musical skills and understanding, I wouldn't confine myself entirely to one area.
People tend to forget that in the old times, among pianists it was almost mandatory to be able to improvise. You were not only judged by your technique, musicality, and ability to play pieces, but by your spontaneous creativity and mastery.
Sadly, somehow improvisation disappeared from the standard skills in the late 19th century, and here we are, admiring jazz players skills.
Had pianists/composers retained improvisation as a core skill, it would have developed into something huge, not limited to one special style or one kind of harmony.
So, learn both. If you start with classical, when you feel ready for it, pick up any kind of improvisational skills - you won't regret it.