Define the word "distinguishable".
Is it: "Differences exist that are discerned only by observation of revealed cards"?
Is it: "Differences exist that are discerned by examination of the hidden cards"? (And do you need to consider just the face down card, or all of the cards in the deck?)
As to which definition you would need to use to properly evaluate probabilities, well, that depends on what probabilities you are required to evaluate.
There are $52$ (equally) possible outcomes for the face up card. If the events of concern only involve that card's value, then that's all your model needs to be concerned about.
There are $52\times 51$ equally possible outcomes for both dealt cards. If the events of concern may involve both cards' values, then that is what your model needs to deal with.
There are $52!$ equally possible ways the deck could have been shuffled. If the events of concern might involve any card's value, then you would have to consider with this. Likely not, though.