The different notations are to a considerable extent outgrowth of slightly different ways of looking at "the derivative." To me *these different ways* is the important thing to discuss,  the notation is a by product. And, I think it is quite useful to spend some time on discussing these different ways and to mention the different notations in that context. (However I understand this is not always possible, see the end.)


For example:

* the notation $f'$ and $\dot{y}$ stresses the (physical) idea of having some function that describes something (some process) and one does something to it to analyze/understand its properties.  

* the notation $D_x f$ emphasizes the fact that one can think of the situation of having an operator (a function!) $D_x$ that assigns to each (under suitable conditions) function $f$ another function $D_xf$. 

* the notation $\frac{df}{dx}$ emphasizes the definition as a limit of the quotients of differences.  

* the notation $df(x)$, in particular when used like $f(x+h) = f(x) + df(x)h + r(h)$ stresses that the derivative at a some point is actually a linear function that just happens to be identifiable with its slope.

There would  also be different ways to split things up and to motivate things.

If one does this the different notations come more naturally and one then can and I think should vary one's own usage and exercises a bit. Still, I would *not insist* or even much encourage students *using* different notations themselves. There is some clear merit in having them *understand* the different notations; for them being able to use different ones (fuently) this is much less clear to me. There is also a small risk that they would use them, but would use them not really 'idiomatically,' as the notations arein some sense not completely interchangable. 

For example even an example in OP seems slightly of to me (though some place else different conventions might be in place): as far as I know, the dot-notation is mainly used by physicist and alike, and does typically not denote just some derivative, but writing $\dot{y}$ "means" derivative by *time*, which "would never" be denoted $x$ but $t$. [For me, as a pure mathematician, the above is strange to write, but I really think that if $y=xt^2$, then to some it is completely obvious that $\dot{y}=2xt$ and not $\dot{y}=t^2$.]

This answer assumes of course that one has time to do this discussion, and implicitly also that the students or a considerable part there of will actually see the higher dimensional theory for example. If this is not the case, I think I would not bother much with different notations. Pick one (based on your own preference or possibly informed by the predominant convention in whatever other field the students might study, if there is one) and stick to it, and just mention once that there are different notations, so that students are not completely surprised when the open some other book. However, if some student prefers to use a different one, accept this, too (as long it is clear and coherent).