However, there is one point of overlap. If you mean a unit of volume frequently used in cooking (250 ml), it is a "cup" in English but a «стакан» in Russian. I am pretty sure it is because the "cup" I use looks just like the glass shown on the picture above. I wonder why is that :)
Стакан is a generic word for a cylindrical glass usually made of glass, a vessel used for non-alcoholic drinks.
There is no other word to call this type of drinking vessel (bar стаканчик). Кружка and чашка have a handle, and бокал/рюмка/фужер/стопка are usually for alcohol (and the shape is different).
As explained above, стакан is a glass, while a cup is чашка. This is a bit complicated though by the fact that the measurement called a "cup" in English is called "стакан" in Russian. It's more common to measure rice in cups, I guess, and nobody drinks rice from a glass, so "стакан риса" is probably "a cup of rice" but "стакан сока" is probably "a glass of juice".
I'll reference a perceptive comment of Shady_arc's in another discussion: one peril of reverse trees is that they sometimes accept shoddy (often read "too literal") versions of the language that their users are (notionally) learning from. That is likely best understood as one such case. Given that "стакан" is more defined by shape and "glass" is more defined by material composition, it seems the way to resolve the inconsistency would be to accept "cup of juice" here.
Americans use cups as a measure for cooking, in the UK people will use a more specific measure of weight or volume. So if a cup of rice is a correct answer, I don't see why à cup of juice isn't.
In the US a cup is not some ambiguous measure. It is precisely defined, and happens to equal 236.588 mL (to which a Russian cup is a pretty close equivalent). To prepare rice one puts together a set volume of dry rice in a known ratio to a set volume of water. I.e. measuring them by cups - the volume measure - makes sense. Such use is not as common for juice, but shows up in things like maximum recommended amounts for a child to consume per day (although more commonly expressed in ounces, i.e. 1/8ths of a cup).
стака́н (stakán) [stɐˈkan] m inan (genitive стака́на, nominative plural стака́ны, genitive plural стака́нов) "drinking glass" From Old East Slavic достаканъ (dostakanŭ), from Turkic (compare Chagatai tostakan (“wooden bowl”), Tatar тустыган (tustığan, “cup”), Bashkir туҫтаҡ (tuθtaq, “cup for drinking koumiss”), Kazakh тостаған (tostağan, “wooden cup”)).