There are issues with how to define $i$ no matter which you choose, as the answers to the linked questions attest.
To begin with, it is not actually true that $\sqrt 4 = \pm 2$. The square root function is defined (by mutual consent of the maths community) to only produce non-negative outputs, so $\sqrt 4 = 2$.
There are two reasons for this. Firstly, a function can't produce two outputs simultaneously or it wouldn't be a function. Secondly, if it did produce two outputs then what on earth would be meant by the notation $\sqrt 2$? If $\sqrt 2$ was two numbers then so would $-\sqrt 2$ and indeed they would have to be the same two numbers so $\sqrt 2 = -\sqrt 2$ which would imply that $\sqrt 2 = 0$.
So $\sqrt 4$ is in fact just 2. If you want to point out that there are two numbers that square to give 4 then you have to say something like "The solutions to $x^2 = 4$ are $x = \pm 2$".
So in that sense, saying that $\sqrt{-1}=i$ does not at all preclude us from saying that the solutions to $x^2=-1$ are $x=\pm i$.
What it does do is implicitly assume that there is a mathematically consistent way to choose which of the two possible solutions is the one meant when you do the square root of a nonpositive number. For example, if you can unambiguously say $\sqrt{-1} =i$, then there ought to be an unambiguous way to define which of the two possible solutions to $x^2 = 1-i$ is the single one meant by $\sqrt{1-i}$. There is a way, but it is not obvious until you talk about polar form, which comes much later!
Defining $i$ using $i^2 = -1$ is problematic too, because if $i$ was a variable then that equation has two solutions, so which of them is $i$? In fact either of them could be $i$ and there is no problem whichever one you choose, as long as you choose one and carry through with it for the rest of the maths you do. If each individual is consistent and doesn't switch between them willy-nilly it will be fine, and it won't matter if your $i$ is different to mine because switching all the $i$'s for $-i$ produces the same set of complex numbers and doesn't interfere with the operations on them. This is the idea of conjugation!
(The resolution of all of this is the representation of complex numbers with points on a 2D plane. Mathematicians agree to choose $i$ to be the point anticlockwise from 1 rather than clockwise.)
I think it would be remiss of you as a teacher not to mention some of this, but on the other hand it would be a bad idea to go into full detail straight up because that would confuse the students. Also, if locally the definition is $i = \sqrt{-1}$ then they need to know this to get the marks in their exam locally.
I would choose the approach of using the definition you have, but pointing out that there are some technical issues you're glossing over by using this definition. I would also tell them that it's possible to define it as a number such that $i^2=-1$, which also has some technical issues. You can possibly get back to these issues later if they are interested.
Of course, if a task they are doing doesn't require them to explicitly mention the definition, then there is absolutely nothing wrong with them using the fact that $i^2 =-1$, which I find is easier than using square roots anyway. Indeed, many students I have taught don't realise it's ok to do this and end up having extra lines in their working and make more errors.
One final thought: whenever there are multiple ways that people define things, it's probably a good idea to let your students know this, and which definition is the preferred one locally. (Another common example is the definition of the number $e$.)
This tells students that it is possible for there to be differences in definition. It also means that they are less likely to be confused by resources they find online which use a different definition.
