You couldn't get a decent T-shirt in the Soviet Union. In my childhood the ones that we had were referred to as майки с короткими рукавами (майка is an undershirt). Then people started using the word футболка that originally only meant a football jersey to refer to T-shirts. In 1990 in the US I heard Russian people say "тишотка" - that was before it was replaced by футболка.
I imagine it became popularized as football jerseys adapted and changed over time during the early 1900s (from long sleeved, collared and thick to short sleeved and thinner, and lighter). This is a complete guess by the way, I'll ask some of my native Russian friends and see if they know more.
I would suggest to translate футболка as "T-shirt", and "рубашка" as "shirt" (exactly as you have done), while also accepting "shirt" as an alternate translation for "T-shirt", since it's often called that in the US.
Then perhaps include a note in the "Clothing" summary, as per explaining short and long coats etc.
Similar to how there's a difference between a landline telephone and a mobile telephone, yet both need to accept "telephone" and "phone"; the former has to also accept "landline", and the latter also has to accept "cell, cellphone, cell-phone, cellular phone, mobile phone, mobile", even though most people most of the time just say "phone" in this century.
In American English at least shirt is used to mean T-Shirt much more often. Rubashka would be referred to as a dress shirt. And it is often specified as such, especially when asking stores if they have it... You ask "do you have a black dress shirt" because if you don't say the dress part, the salesperson will think you mean T shirt and say yes, even though you're looking for a rubashka which they might not have.
So in my opinion, you've made the opposite of the correct choice so far. Leave t-shirt as the unspecified shirt, and rubashka as the specified dress shirt. It matches up to my expectations when I was learning Russian in Russia as well.
I agree and disagree with "Shirt and t-shirt are one and the same." In American English, a t-shirt is a lightweight, short-sleeved shirt that you would wear in informal settings. They often have words or images printed on them and almost always have a simple rounded neckline. "Shirt" is a generic term that includes t-shirts, jerseys, polo shirts, and button-down shirts. In speech, however, we often don't specify. If I wanted someone to put on a button-down shirt, I'd probably just say, "Put on a nice shirt." Department stores often call them "men's long-sleeved shirts," or "dress shirts." Then there are blouses. These are fancier women's shirts, either a button-down shirt or something dressy, long-sleeved, and flowing. Blouses are not tight-fitting.
Another note: a dress shirt in British English does not simply mean a long-sleeved shirt. It is one suitable for wearing with evening dress (either black-tie or white-tie). As such, it will have holes for cufflinks at the wrists, rather than buttons, and may well have a wing collar - indeed, both collars and cuffs may be detachable. Such shirts often have ornamentation such as ruffles, down the front. They are for wearing with tails, maybe an evening jacket, but never alone or with a suit.
Referring to a T-shirt as a shirt would cause confusion here (in Britain).
If one says "shirt" in Britain, one definitely always means the kind that buttons together, has a collar, etc (unless otherwise specified; ie, by saying "T-shirt", "polo shirt", etc).
That said, I'd agree that the American English "shirt" should probably also be accepted (since Americans do call it that), as perhaps should the fuller literal translation of "football shirt", assuming that a football shirt is indeed still also called "футболка".
I suppose it's because I'm ancient, but to me "shirt" and "T-shirt" are not one in the same. "Shirt" is a generic that covers polo shirts, dress shirts, Hawaiian shirts, Mexican wedding shirts, hair shirts, and T-shirts, among others. "T-shirt" refers to one specific type of "shirt."
Edit: I'm guilty of writing a response before reading all previous responses; forgive me, jairapetyan, for seemingly plagiarizing your comprehensive comment. I would add, however, that the few and proud United States Marines wear a blouse—not a shirt—that is crisply starched and pressed so as to present a fitting background for ribbons (or medals, depending on the uniform of the day) that represent incidents from their service.