To begin with, a website designer is responsible for typography and needs to take care of fonts, line length and kerning. He/she may also be expected to avoid complications, which might arise if fonts other than the ones considered safe for browsers, are used. Chances are that the responsibilities include the whole of visual aspect and not just a part of it, typography that is. Simply put, it is the designer who must decide on the layout and think of a suitable colour scheme.
The standards here may be related to the layout of the page or might have something to do with the coded elements, depending entirely on the situation. For that matter, the standards must also be met in terms of readability. That pretty much sums up what a designer does, of course in addition to ensuring that the design turns out to be a user experience one.