What is a "vocal coach" and how does it differ or is specific in comparison to the vocal learning / training in general?
I worked as a vocal coach for 20 years, mainly working with actors and preparing them for musical theatre and auditions. I always made it clear that I was not a singing teacher. In other words, I never taught technique. Here are some examples of my input:
Where to breath in a song, finding the correct key for their range, presentation of the song: this often involved finding the character... where to look, when to move, how to stand, what to wear, which accent to adopt, editing the song for their auditions. I would play the accompaniment and mark the music for repeats, endings etc.
I always worked in conjunction with a singing teacher. A vocal coach needs to be a proficient musician and a decent piano accompanist, whereas a singing teacher is always, primarily a trained singer. http://www.learnjazzpianoonline.com
A vocal coach helps you learn techniques to strengthen voice and breath, and extend range etc. They also identify bad habits and help you fix them.
They do carry out the tasks you describe as vocal learning.