Can I call a model wherein Bayes' Theorem is used a "Bayesian model"?

No

I am afraid such a definition might be too broad.

You are right. Bayes' theorem is a legitimate relation between marginal event probabilities and conditional probabilities. It holds regardless of your interpretation of probability.

So what exactly is a Bayesian model?

If you're using prior and posterior concepts anywhere in your exposition or interpretation, then you're good to call your model Bayesian.

In a broader sense though you must be subscribing to Bayesian interpretation of probability. This little theorem of Bayes was extended and stretched by some people into this entire world view and even, shall I say, philosophy. If you belong to this camp then you are Bayesian. Bayes had no idea this would happen to his theorem. He'd be horrified, me thinks.

Can I call a model wherein Bayes' Theorem is used a "Bayesian model"?

No

I am afraid such a definition might be too broad.

You are right. Bayes' theorem is a legitimate relation between marginal event probabilities and conditional probabilities. It holds regardless of your interpretation of probability.

So what exactly is a Bayesian model?

If you're using prior and posterior concepts anywhere in your exposition or interpretation, then you're likely to be using model Bayesian, but this is not the absolute rule, because these concepts are also used in non-Bayesian approaches.

In a broader sense though you must be subscribing to Bayesian interpretation of probability as a subjective belief. This little theorem of Bayes was extended and stretched by some people into this entire world view and even, shall I say, philosophy. If you belong to this camp then you are Bayesian. Bayes had no idea this would happen to his theorem. He'd be horrified, me thinks.