What exactly is a Bayesian model? Can I call a model wherein Bayes' Theorem is used a "Bayesian model"? I am afraid such a definition might be too broad.
So what exactly is a Bayesian model?
 A: 
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.
A: A statistical model can be seen as a procedure/story describing how some data came to be. A Bayesian model is a statistical model where you use probability to represent all uncertainty within the model, both the uncertainty regarding the output but also the uncertainty regarding the input (aka parameters) to the model. The whole prior/posterior/Bayes theorem thing follows on this, but in my opinion, using probability for everything is what makes it Bayesian (and indeed a better word would perhaps just be something like probabilistic model).
That means that most other statistical models can be "cast into" a Bayesian model by modifying them to be using probability everywhere. This is especially true for models that rely on maximum likelihood, as maximum likelihood model fitting is a strict subset to Bayesian model fitting.  
A: Your question is more on the semantic side: when can I call a model "Bayesian"?
Drawing conclusions from this excellent paper:

Fienberg, S. E. (2006). When did bayesian inference become "bayesian"? Bayesian Analysis,
1(1):1-40.

there are 2 answers:

*

*Your model is first Bayesian if it uses Bayes' rule (that's the "algorithm").

*More broadly, if you infer (hidden) causes from a generative model of your system, then you are Bayesian (that's the "function").

Surprisingly, the "Bayesian models"  terminology that is used all over the field only settled down around the 60s. There are many things to learn about machine learning just by looking at its history!
A: In essence, one where inference is based on using Bayes theorem to obtain a posterior distribution for a quantity or quantities of interest form some model (such as parameter values) based on some prior distribution for the relevant unknown parameters and the likelihood from the model.
i.e. from a distributional model of some form, $f(X_i|\mathbf{\theta})$, and a prior $p(\mathbf{\theta})$, someone might seek to obtain the posterior $p(\mathbf{\theta}|\mathbf{X})$.
A simple example of a Bayesian model is discussed in this question, and in the comments of this one - Bayesian linear regression, discussed in more detail in Wikipedia here. Searches turn up discussions of a number of Bayesian models here. 
But there are other things one might try to do with a Bayesian analysis besides merely fit a model - see, for example, Bayesian decision theory.  
A: A Bayesian model is just a model that draws its inferences from the posterior distribution, i.e. utilizes a prior distribution and a likelihood which are related by Bayes' theorem. 
