It can't be that important because I have never heard of it before (and teach at university level). I believe textbooks will have some things because they provide an easy form of question rather than because they actually matter.
Edit: The more I think about this, the more I agree with your teacher, in that I think it is unhelpful to teach both given that only one is needed. What I mean is, the 'point-slope' equation is the same equation as the 'slope-intercept' equation. As such, I think it is unhelpful to teach one concept as if it is two different ones. I believe that telling students incorrectly that these are different will encourage their natural tendency to think of the equation of a line as a string of characters, rather than a mathematical statement that is true for points on the line and false otherwise.
Also, emphasising the difference makes out that there is an important difference, when all the difference is is basic algebra. I would think this sends the message that the algebra is difficult and should be expected to be difficult (already far too wide-spread an opinion). Instead we should be teaching that the algebra is not difficult and is not the important feature.
My suggestion, then, would be to teach both ways of writing the equation, showing that one or other may be more convenient depending on the information you have available, and can help in different ways to aid intuition. Emphasise that the two only differ by rearranging the formula, and that they can easily move between the two to get the viewpoint they find helpful. Also make it clear that mathematical convention is to write $y=mx+c$, and so they should give their answer in this form, whichever way they arrive at it. (Before I get objections to this last point: if a student wrote something else I would consider it laziness, and if I saw anything else in a paper without reason I would be very surprised.)