In a transformer, how do you determine where the induced EMFs (E) and voltages (V) are placed in a vector diagram?
The question really seems too long. Anyway ... about this excerpt:

My question is why are those induced EMFs selected as the reference for those cases? Why not the voltages (V)? It might be for convenience or tradition, but I was wondering if there are other reasons besides those if those are one of the reasons.

Using EMF as a reference facilitates the understanding of the influence of the Power Factor on voltage drops and helps to establish the fact that it is only there that the transformer's turns ratio is fully satisfied, regardless of whether the transformer is in no-load or under load.
And about this:

Then for Cases 1 and 2, why are E1 and E2 in phase with each other and why on the negative y-axis?

I believe it was for convenience, to avoid many overwritten vectors. The fact that the polarity (direction in which the coil was wound) is positive or negative will cause these two vectors (E1 and E2) to have the same or opposite direction.