Sounds like you get the left and middle bits. I’m not even sure this is the intention of the meme’s creator, but my take follows.

First off, it’s true that both 2 and -2 are solutions to the equasion x^2 = 4. And it’s true that the first time you run across the square root operation in math, they’ll probably describe it in terms of solutions to the aforementioned equasion. Or perhaps more simply as “the opposite of squaring.”

And whether the square root (strictly speaking) is or is not straight defined in terms of whether x^2 = y, there’s definitely a strong relationship between the two.

However, we have a choice whether we define that √ symbol as “all solutions for x to equasions of the form x^2 = y” or “the positive solutions for x to equasions of the form x^2 = y” (or for that matter, something else entirely, but we’ll pretend it’s just these two for now.)

So, the next question is “which definition is more useful?”

Consider the Pythagorean theorem. c^2 = a^2 + b^2. If we know, for instance, that a and b are 1 and 1 respectively and we want to know c, we can solve for it easily, but we have a choice in how to represent it. We could say c = √(1^2+1+2) = √2 and interpret that √ as implying both the positive and negative solution. Or, we can say √ just meant the positive one and write c = ±√(1^2+1^2) = ±√2.

For this specific case, solving for the length of the hypotenuse of a right triangle with the other two side lengths being 1, it kindof doesn’t make sense for lengths of sides of a triangle to be negative, right? Would you say “the lenght of the hypotenuse could be the negative square root of two or the positive square root of two?” Probably not, right? So we at least have one good example where it makes sense to even discard the negative result when taking a square root.

Another situation might be different, though. Maybe you’re trying to find the roots of a quadratic function using the quadratic formula. Note that the quadratic formula is almost always explicitly written with the ± to explicitly indicate that both the positive and negative value are (at least potentially) useful. In that case, it’s beneficial to have both the positive and negative roots.

So, in short, it feels like there are some cases where it’s useful to drop the negative result and others where it’s not. If we define √ to include both, we don’t really have the freedom to choose. (Sorry. Every trig student from now on must specify that the length of hypotenuses can be negative or positive. But then why stop there? That almost implies that every length/distance is both negative and positive. But that really serves no one.) But if we define √ to only be the positive one, we can choose at the point at which we write down the √ symbol.

We can still say "solving x^2 = 4 for x, we get x = ±√2. And we can still say “the length of the hypotenuse of a right triangle with the other two side lengths being one is √2” and not have to introduce a lot of unnecessary pedantic overhead. And every time someone does math, they can make the decision at the time they write down the √ whether to write √ or ±√.

That’s most of it, but just to tack on another couple of reasons, it’s kindof just more convenient to deal with equasions that have (at most) one “output” when you can get away with it. And as a software engineer, it’s more convenient to have the sqrt(...) function only return the positive result. (I do a lot of CAD work with OpenSCAD and I’m glad the sqrt(...) function only returns the positive result there. If I ever did need the negative result as well, it’s simple enough to deal with that without having functions that return multiple values.)

So, in short, the naive way of defining “square root” is in terms of "solutions to formulas like x^2 = y, but in practice it’s a lot more convenient if we define “square root” in terms of “just the positive solution to formulas like x^2 = y” and have the flexibility to still write the ± when we get benefit from it.

And that’s just my take.