I cannot think of any difference with or without the use of the definite article in this sentence. The fact that water is an uncountable noun may have influenced me in some way.
I don't want to drink (the) water from this cup.
Is there any difference?
The article does change the meaning.
I don't want to drink the water from this cup.
This is about the specific water that is currently in this cup. If the cup is empty it wouldn't make sense to say this. Perhaps the water is stale or there is mud in the water. But if you replace the water with some fresh water then I would drink it. Normally you would say "I don't want to drink this water".
I don't want to drink water from this cup.
This is about any water that you could put in the cup. It doesn't matter if there is water in the cup now or not. But I might drink something else... perhaps if you put vodka in the cup I'd drink it! Normally you would say "I don't want to drink water".
I don't want to drink water from this cup.
This is about this cup. There is something wrong with the cup, perhaps it is cracked. But if you get me a new cup and fill it with water, perhaps I'd drink that. Normally you'd say "I don't want to drink from this cup".
It could also mean "I would drink something if you changed both the water and the cup"
Without any emphasis there is a slight ambiguity, that would normally be resolved by the wider context. The "normal" sentences are better as they avoid this slight ambiguity.
I did very quick research and found that water is used without the definite article in many cases. Check those two examples below:
1- https://science.nasa.gov/science-news/science-at-nasa/2013/01nov_ismsounds
(Without air, there is no sound)
2- https://www.businessinsider.com/how-many-days-can-you-survive-without-water-2014-5?r=US&IR=T
(A man drinks water near a fountain on a hot summer day in Brussels, Belgium, July 19, 2016)
Resource: Without (the / no article) air and water, living things could not survive
