A 'glass holder' is extremely nebulous because it could variously mean a 'holder' made out of glass or could be a standing base [ein Sockel] to hold some kind of drinking glass*, drinking or otherwise that stands on a countertop, etc.
A bracket is commonly understood to be a wall-mounted holder or support of some type.
For example, a wall-mounted shelf or a curtain rod is mounted to the wall by means of brackets.
A rack is often longer than a bracket and meant to hold one or multiple items such as screwdrivers or garden tools for easy reach on a wall –OR to suspend and stow items such as bicycles, skis, etc. in a garage or on the rear or top a vehicle, for instance.
*In modern AE (at least) a beaker is understood to be a glass, plastic or metal container (in der Größe eines Bechers) for use in a laboratory, not in the home.
A tumbler is usually a large 'drinking glass' (meaning also plastic or metal), somewhat larger, I'd say, than the typical glass or plastic cup one would keep in the bathroom for mouth rinsing, etc.
Since AE does not use the perfectly descriptive term 'beaker' for drinking containers that one puts one's lips to, it leaves the language somewhat in flux about what to call various types of these. As plastic and paper have replaced glass to such a degree, the term cup or drinking cup has become more common as a general term for all such items without handles, whereas cup or mug are the only terms in current use for such items having a handle.
–It is still very common, however, to ask for, or to offer someone 'a glass of water'. There is no implied expectation as to how it will arrive. Usually people will ask for 'a cup of water' in a situation where actual 'glass' is unthinkable, such as at a fast food counter. In a restaurant, on the other hand, I would ask the server for 'a glass of water' with no hesitation.