In the uncertainty principle, are $\Delta x$ and $\Delta p$ uncertainties? [duplicate]

Question

I was thinking about a very old exam where the UP was expressed as $$\Delta x\Delta p\geq\frac{\hbar}{2}.$$

From what I understood in my measurement course, the uncertainty of a measurement model $Y$, $u$, is written as $\delta_{Y}$ in the case of its systematic uncertainty and $\sigma_Y$ for the statistical uncertainty. Either way, the associated error of measurement is expressed as $\Delta_Y$.

Now, I am left with some doubts in my minds:

1. I read that the HUP is associated to the intrinsic uncertainty of the system and not to measurement uncertainties: perhaps the physical notation for intrinsic uncertainties is different?
2. What is $\Delta$ in $\Delta p$, an operator? If that's the case, perhaps the uncertainty is given by $\Delta$ applied to $p$?
3. If it's uncertainty, then the system can assume values of $x$ and $p$ such that the product of their errors is less than $\frac{\hbar}{2}$ (thus the possibility of a more accurate measurement than whatever "limit" the HUP provides), as long as they're within an interval that satisfies $\Delta x\Delta p\leq\frac{\hbar}{2}$? In introductory courses, it's often described as if it's impossible to measure their (well-approximated) value, rather than not being able to tell if it's (reasonably close to) the "true" value; perhaps that's where my doubts come from.
4. I understood nothing and I should revisit everything from scratch (very possible scenario).

I haven't checked other definitions of the UP, its relations and whatsoever, so with this definition I am really confused and I don't get what I got wrong.