I would agree with this.
If I was being specific then I'd call the top in the OP a poloshirt (and not a shirt) but if I was being generic, then it would be a top or t-shirt.
If I was told to picture a "t-shirt" then I'd think of something without a collar, but if for example I was placing children's clothing in a drawer, I put poloshirts and t-shirts together and call them all t-shirts for brevity. I tend to think of "top" or "t-shirt" as being the generic category of garments that go directly next to your skin unless it's cold enough to also wear a vest (which I think Americans call undershirt). Unless otherwise specified, a t-shirt is short sleeved and no collar. But if I was not being specific then a long sleeved top and poloshirt could both be referred to as "your t-shirt" or "pass me that t-shirt please" or "pack four t-shirts" or "DC2 is running out of t-shirts, can you put a wash on?" (I might also use "top" in place of any of these)
Just like everything that goes over the top of the t-shirt is a jumper. Even though my kids mainly have sweaters and hoodies, and nary a knitted item in sight.
And weirdly, the word "shirt" when used on its own definitely means a shirt with buttons and a collar to me. I get the sense from American parenting sites that they use "shirt" as we use "top".