This is a very vague area, and different grammars lump the different types of expression into different categories - none of which really work. The mistake is thinking of partitive and quantifying as grammatical categories at all. If I can be boring and repeat what I've said before - grammar is meaning. Partitive and quantifying are semantic categories or "meanings". A word may have a partitive use, a quantifying use or both. But this doesn't mean that it "is" partitive or quantifying. Some examples :
The definition of a partitive is that it's a word which expresses part/whole relationship -
a piece of cake, a bit of cake, some of the cake, a lot of the cake, all of the cake - all these can be contrasted or equated with "the whole cake"
Thus words usually described as quantifiers may be partitive in meaning - but they may not. If I say
I ate some cake or
I ate a lot of cake, I am simply talking about the quantity of cake that I ate, not relating it to a whole cake.
If you stop trying to divide the words into grammatical classes (ie saying a word must be either X or Y) and just look at the meaning they express, it becomes a lot clearer.
Expressions like
a cup of tea, a bottle of wine etc, on the other hand, are different. The nouns
cup and
bottle don't refer to parts of the whole at all and therefore don't seem to me to be partitives at all. They refer to different objects. If I say
I bought a bottle of wine, (as I frequently do
) I'm not referring to a part of the wine, nor necessarily to the quantity but to a specific object which just happens to contain something else.
And similarly I wouldn't class expressions such as
a blade of grass,
a grain of sand or
a grain of rice as either partitive or quantifying either. The Communicative Grammar describes them as "unit nouns", used with objects which the language usually considers mass but which are actually distinguishable objects. And again some words, like
bit, may express all three meanings - unit, partitive and quantifying.
Can you you pass me a bit/piece/sheet of paper? = unit meaning
Can I have a bit/piece/slice of that cake? = partitive meaning
I've got a bit of a / a slight headache = quantity meaning
So my answer would be that far from lumping all these words together we should actually not think about grammatical category but look at a far wider range of meanings* that they may express.
* There are actually more than just these three, but I think three are enough for now ....