I don't think that they are choosing which model to give to the customers, as much as they are choosing which model seems best for managing risk that their cardholders pose to themselves. (This would be the soft pull they do monthly on existing cardholders.) They make that choice based on a lot of things: the cost of the model (FICO charges them for pulling it), the cost of the bureau, the company's previous experience with the same or similar model, etc.
Once they have made that choice, then the score they provide to the customers is whatever that score was. E.g. Citi monitors its customers with EQ FICO Bankcard Enhanced -- so Citi cardholders get that score. Amex uses EX FICO 8 Classic. And so on.
This is all part of that program I mentioned earlier in this thread: FICO Open Access:
http://www.fico.com/en/products/fico-score-open-access
It was started in Nov 2013.
There are a few places where you can get a free FICO 8 score without having an account with a particular bank or CC issuer. For example, creditscore.com. I assume that their business model must be based on advertising or selling your interest in FICO scores to CC issuers, who in turn market their products to you.
But in general the free FICO scores are almost all given to customers as a consequence of what the company is using to assess risk to itself.
