The words you are looking for are not “countable” and “uncountable” but discrete (countable) vs. continuous (infinitely divisible) quantities.
Did I use those exact phrase sin a perfectly correct English sentence.
Why, I believe I did.
So I win. 
Wins the thread.
It’s all about context. There is no one answer to satisfy every situation.
I gleaned this distinction fairly soon in the thread, and wondered if I was going to get to post it myself.
When you puree the peaches, you get a continuous glob of them. Before they were in discrete units. A huge peach might be “a lot of peach” because below the single unit of “one peach” there’s the continuous fruit of the peach that cannot be counted.
Using “counted” there makes me wonder: the atoms/cells of the peach can be counted (assuming peachness extends to cellular level), just not on a human scale. Is there ever a case where you would use a different word for a case where there is actually an infinite number of something? What about the difference between countable and uncountable infinities?
Ok, that last paragraph was silly.
You could have seven lots of rashers of bacon.
Or seven too few lots of rashers of bacon…
It seems to me that the complexity here is a matter of metaphysics (specifically, ontology), the basic structure of reality, not of language. As such, the Japanese language must surely have a way of dealing with these issues too. How does Japanese deal with the difference between having too many peaches, and having too much peach in your pie? Once you know that, it ought not to be too hard to explain toJapanese people how English copes with it.
They’re asking you about Classifiers. They’re confused because Japanese uses classifiers and English very rarely does. So start your lesson off with a discussion of that. Explain that English doesn’t use a strict classifier system and these questions are a relic of that fact.
In japanese, you wouldn’t say “I have two cakes”, you’d literally have to say “I have two cake-like objects of cake”. Look here for another example.
So they’re used to saying, essentially, “there are two small animals of cat”, but in English, you’d actually say “There are two cats”. That seems to be where the confusion is coming from. To them, all things are quantifiable. They’ve been saying their whole lives, basically, things like “there’s too much cat”.
Explain that there’s no hiki word in English and maybe that’ll set of the lightbulb.
Missed my edit:
Here are some ways to phrase it better than my hackneyed job in my previous post:
In Japanese, virtually all nouns must use a counter to express number. In this sense, virtually all Japanese nouns are mass nouns. Mass nouns (also uncountable noun or non-count noun) are is a common noun that presents entities as an unbounded mass. In English, these words must be combined with a unit of measure (e.g. liters of water), whereas countable nouns can take an article (a, an, the).
Some words are both mass nouns and count nouns, like paper, food, cake, etc. To see the difference, read Count noun - Wikipedia.
The OP and ensuing discussion reminds me of my sophomore year in college, when I had a roommate from New Hampshire. If we were out of beer, he’d say, “Let’s go get some beers.” If it had been me, and it had come to my attention that we were out of beer, I’d say, “Let’s go get some beer.”
I’d never heard of “beer” being pluralized in that sense before. It’s certainly not incorrect, but it sounds a little strange to me. I wondered, and continue to do so (because I often find myself in a need for more beer), if it’s a regionalism.