Sounds right - According to what's on the UOGamecodes site,
"# One charge will be consumed when you claim your pet.
* You can restrict users by access level.
# All players who can enter the house can use the "stable" command freely and it consumes no charges."
So if your house is public, I think that would mean anyone can stable, but only whoever you define as being allowed to by the security settings you apply to the post can claim.