There might be some differences between British and US English on this one, but a 'dress shirt' in British English is a very formal shirt - the kind you would wear with a tuxedo. These do not normally come in short-sleeved variants.
The other kinds of fully-buttoning shirts (ie not a 't-shirt' or a 'polo shirt') may be described as a 'formal shirt' (worn for 'smart', but not 'black tie' occasions) or a 'casual shirt' (for casual wear, usually without any kind of tie). Often, a 'formal' shirt may simply be called a 'shirt', because if it isn't dress or casual then it doesn't need further definition.
Both formal and casual shirts may come with short sleeves or long sleeves, so the only way to specify what you mean would be to say "a short-sleeved shirt" or "a short-sleeved casual shirt".
The kind of shirt shown in your picture is most definitely casual - worn without a tie, top button open, and with a designer logo showing. This is not formal, and certainly not 'dress' wear.