Remember I did say the phrasing wasn't rules-lawyer proof.
But if you wish to play, I'm game. Please respond to the following counter-points:
1) your PHB quote talks about check results. It does not even mention rolls in this context. What are your arguments why this quote should be applicable?
2) as a budding rules-lawyer, you should know that bringing up other, unrelated, rules passages provides circumstantial evidence at best. In other words, there is no reason to believe the designer of Split the Tree was making a point in the "check roll result" debate. Besides, how do I know you - in your search for an example that supports your POV - didn't skip examples that does not; or indeed examples that supports the opposite position?
3) assume for the moment you are wrong and the feat is intended to share the roll only and not the entire check result. How would you phrase this in any other way than "both of you can use the higher result of your two rolls"?
To repeat, my view is that any ordinary gamer should go with the "shared roll" interpretation over the "shared check" one (unless clarification/errata becomes available, of course), while rules-lawyers can't be stopped from going with any interpretation they fancy.