Appropriate Notation: $\equiv$ versus $:=$

With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in

$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$

which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's blog :

$$S(x, \alpha):= \sum_{p\le x} e(\alpha p) $$

Is it user-background dependent, or are there certain circumstances in which one is more appropriate than the other?

An "equality by definition" is a directed mental operation, so it is nonsymmetric to begin with. It's only natural to express such an equality by a nonsymmetric symbol such as $:=\, .\ $ Seeing a formula like $e=\lim_{n\to\infty}\left(1+{1\over n}\right)^n$ for the first time a naive reader would look for an $e$ on the foregoing pages in the hope that it would then become immediately clear why such a formula should be true.

On the other hand, symbols like $=$, $\equiv$, $\sim$ and the like stand for symmetric relations between predeclared mathematical objects or variables. The symbol $\equiv$ is used , e.g., in elementary number theory for a "weakened" equality (equality modulo some given $m$), and in analysis for a "universally valid" equality: An "identity" like $\cos^2 x+\sin^2 x\equiv1$ is not meant to define a solution set (like $x^2-5x+6=0$); instead, it is expressing the idea of "equal for all $x$ under discussion".

The notation $x:= y$ is preferred as $\equiv$ has another meaning in modular arithmetic (though it is almost always clear from context as to which is meant). However, there is one big advantage to using the $:=$. That is, it is not graphically symmetric and hence allows for strings such as $$ y:= f(x) \leq g(x) =: L $$ where here we are defining both $y$ and $L$. This statement would be much more cumbersome using $\equiv$, and it would not make sense if one simply wrote $$ y \equiv f(x) \leq g(x) \equiv L. $$

$x:=y$ means $x$ is defined to be $y$.

The notation $\equiv$ is also (sometimes) used to mean that, but it also have other uses such as $4\equiv0$ (mod 2).

I encountered $:=$ a lot more than $\equiv$ , and it is my personal favourite.

There is also the notation $\overset{\Delta}{=}$ to mean "equal by definition"

By the way, some people also use the notaion $x=:y$ to mean $y$ is defined to be $x$

It's entirely up to the whim of the author. Other symbols that can mean the same thing are $\triangleq$ and $=_{def}$. I think that only a minority of authors use any special notation, however; the majority just use a regular equals sign.