Did the Roman citizens of Britain feel a different sense of pride for the land they lived on/came from than say, a Roman citizen of Judea? Or was it all universally Roman? I understand that different religions were practiced, but were there regional differences that resulted in varied aspects of Roman culture as well?
Yes, absolutely. Cicero probably phrased it most succinctly in de Legibus: every Roman has two homelands, one of nature and one of law.
Rome is the homeland of law; that is where the Roman derives his citizenship and where he owes his duty. The homeland of nature is where he was born, and where he owes his affection.
The surviving writers of the Augustan period make mention of their homelands and try to call attention to them. Livy mentioned Padua as an alternative to Rome as a center of empire, if history had happened differently. Vergil and Propertius claim to be the Homer of Mantua and Assisi respectively. Ovid, as infatuated as he is with Rome, still makes mention of Sulmo. People like Velleius Paterculus and Asinius Pollio remember their rural origins with pride. Claudius was a special patron of Lugdunum where he was born. It's all over the place if you look for it.
There are also biases against "Romans" from the wrong part of the empire. Claudius, as I mentioned, was born in Lugdunum, and that is a point of mockery in the satire of his afterlife, the Apocolocyntosis. Ausonius either knew or invented a citizen from Britain named Bonus, and wrote several poems pointing out the fact that the only good (bonus) Brit was a dead Brit. Gibbon's judgment on this poet (probably the only time I'll ever agree with Gibbon) was "the poetical fame of Ausonius condemns the taste of his age." #https://en.wikipedia.org/wiki/List_of_burn_centers_in_the_United_States
Romans had as much attachment to their home towns as we do. Being American doesn't mean you have some emotional connection to Washington or New York; we shouldn't expect it was different for them, and it wasn't by their own reports. And they were perfectly capable of holding biases against other Romans for being from the wrong place.