Given Tullius' opinion and understanding of Skyrim and Nordic culture in general, he probably trusts them as far as he can throw them.
I'd say that Tullius trusts the Nords fairly well, it's just that he doesn't really want to be in Skyrim trying to figure out thier ancient traditions, at all. He just wants to kill Ulfric, smash the Stormcloaks (killing Ulfric does this anyway), and go home.
Originally Posted by Calemyr
You were caught in the middle of an ambush. You are at best in Skyrim illegally and at worst an agent of the Stormcloaks. Tulius didn't make the call on your execution, he allowed the Captain in charge manage things, but he doesn't hold it against you when/if he sees you next. The captain is an uncomplicated woman who would rather apologize for executing a minor criminal than for allowing a Stormcloak agent go free under the guise of an innocent bystander. To a certain sort of mind, that would be a logical conclusion. From a personal perspective, it's less forgivable.
Pretty much this. Tullius didn't really care about you one way or the other, and you've been caught hopping the boarder, or at least within ambushing range of the Stormcloaks.
Rowe's Rule: the odds are five to six that the light at the end of the tunnel is the headlight of an oncoming train.