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I also find bending the string by pulling it towards the floor (vs. pushing it towards the sky) is easier to control and get a smooth transition, particularly for the wrist motion described above.
=== Width ===
Width refers to how far the peak frequencies deviate from the original pitch. A wide vibrato has a large deviation.
If the vibrato is too narrow, it's hard to tell it's even there. There may some cases where this is desirable, but for the most part why bother? If the vibrato is too wide, it will sound chaotic, deviating too far removed from the original pitch. In other words, the vibrato will become its own point of interest, rather than simply embellishing the original note.
Again, practice slowly, trying to peak at around a 1/4 bend. If your vibrato is too wide, try practicing making it as narrow as possible while still having smooth transitions. If it's too subtle, do the opposite. Basically, cage yourself in your weak areas - you are not free to leave until you have defeated the challenge.
=== Speed ===