Yes, Casters know about spell slots.
TL;DR: Casters have all the information necessary in-game to figure out that spell slots are a real thing, something which can they can investigate and count, and something that affects their day-to-day lives as spellcasters.
In Liz's case, "after years of apprenticeship and countless hours of study" she has learned to cast spells!
She knows three cantrips which she can cast at will. She has a spellbook which contains six spells. She can spend time each morning memorizing some of her spells. However many she can memorize, it's the same, one day to the next.
She can cast one or more of those memorized spells two times in a day.
(Her players writes a 1 in the box next to level.)
Life is good!
Time passes. Liz has some adventures, gains some experience.
One day, perhaps after additional study, she finds she can memorize one more spell and not only that, she can cast three spells a day! And, she's been able to add to the spells in her book. The magic is strong with this one!
(And, of course, her player crosses out the 1 next to Level and writes 2. Yay!)
Again time passes. Liz has some adventures, gains some experience.
Again one day, she can memorize one more spell and not only that, she can cast a whooping six spells in a day! Plus she's been able to add more spells to her book. Truly she is a force to be reckoned with!
(And, of course, her player crosses out the 2 next to Level and writes 3. Whoopee!)
Now Liz notices something interesting. She can cast magic missile six times in a day. Or she can cast magic missile four times and darkness twice. Or, and this blows her mind, she can cast magic missile twice but she can add some oomph to it and get four missiles. But when she does that, each time she does a four-missile magic missile, she gets once less cast of darkness that day.
Now, Liz is pretty smart. She certainly has the smarts to figure this out. And she's motivated. After all, how many magic missiles in a day can be literally a life-and-death question. But maybe she isn't all that detail-oriented, and she doesn't put it all together. But it is all there. Spell slots are real things to her which are discoverable and countable.
Maybe she doesn't call them slots. Maybe she calls them points, or pips, or meters, or quizzles. But it's entirely reasonable that she says, "Hey, Rogue, I only got two quizzles left! I can cast some darkness, and that uses them both, or I can hit 'em once with a magic missile and still have a quizzle left. What do you want to do?!"
Of course, no table has to play this way. You could assume that somehow how many spells Liz can cast in a day is not invariant, and that The Wizard Table is just an average or an abstraction.
But if at your table it makes sense for Liz to know she's got slots, you can go ahead and let her know that, without fear of meta or OOC.
Liz thinks to herself, hmmm, obviously some spells are more powerful than others, some take one quizzle, some take two. It's almost like I could group spells together into . . . levels. No, tiers is better. Spell tiers. Interesting. Wow, if I could only learn a spell of the third tier. Perhaps some day I'll reach another level, hmm no that's not a good word, plateau is better, yes plateau of power. Yes, maybe some day I will reach another plateau of power and learn a new tier of spells. There's that fireball thing I've been working on . . . .