Why would he lie to caitlyn though?
I'm currently reading Book 1 & I wonder the same thing. Catelyn is straight-up abusive to Jon Snow and it looks like Sansa follows in her footsteps by looking down at Jon & only acknowledging him as her b*stard half-brother.
For the most part Robb, Arya, Bran, & RIckon acknowledge Jon Snow as their brother without using qualifiers like b*stard & half-brother.