Good question, anyways it would be equivalent as a sine and a cosine are the same function with a constant phase difference. 

I always think about it as a pure choice of reference point. You start from equilibrium (y(x,t)=0) so you model the wave as a sinusoidal-shape function.

Because cosinusoidal has one more syllable than sinusoidal.

Very good question. non of responders have noticed essential part of the question:  harmonic signal is described as cos(wt+phi) and not otherwise. Therefore such wave should be called cosinusoidal.

Most probably the reason is that we always draw it from 0 amplitude, which is actually sinusoid.

I would like to draw your attention that signals that repeat are periodic signals, sinusoid is just a small fraction of that type of signals. In case it is sin or cos function of ONE frequency it is called harmonic signal.

Anna Maria is correct. In most Electromagnetics textbook reference is make to cosinusoidal signals, if you would use the sinusoidal signal for reference you would simply change the phase. Of course one must know which is the signal reference since when you see the corresponding phasor you have no track of the cosinuisoidal/sinusoidal behaviour in time-domain.

The name is used to describe  the shape of the function! Cosine and sine are identical in shape. Then it is logical to use the shorter form: sine.

Your question is the same question that asks: "the chicken or the egg came first ?"

Everyone answered: Cosine and sine are identical but to their phase.