There have been many articles published so I encourage you to google and search and find more details as these distinctions need to be clear in a singer's head so you know what you are looking for and use the terminology correctly.
[I am perhaps in a rather unique position as I work and have worked as both teacher AND coach, sometimes with the same singers and sometimes one or the other. I work very hard to keep the distinction clear and make it clear to the singer so the boundaries are not blurred. I would like to add that I primarily work as voice teacher and do dramatic coaching - and encourage my singers to find voice coaches who will work specific details and have specialities in certain repertoire/languages - to give a more thorough and rounded balance to their studies.]
These are my definitions - and cover ALL genres!!
How I would define voice TEACHER: this is someone who understands and works with the physical, psychological, emotional, spiritual nature of the voice and how to build it. The primary focus is the VOICE. Working on creating and building an instrument and a technique that will serve the singer, generally and specifically. I believe teachers need to sing/or have sung - and need to be able to demonstrate thoroughly in order to fully access a singer's ability to learn and absorb information.
Voice teachers and voice lessons are really primarily about the VOICE - developing it, building it, maintaining it, and working on the technical and physical behaviors to find the best possible instrument a singer has! Many voice teachers do not play piano, and so singers often work with a pianist in studio on the technical adjustments in repertoire as well. Some of us DO play and pianists aren't always necessary during a lesson. Voice teachers can work on interpretive/musical ideas as well, but their PRIMARY concern is about the development of the VOICE and work often on the technical aspects of getting a piece of music into the voice.
Voice COACHES work with "all the other stuff" in more focus!!! They coach the MATERIAL from the piano, musically, stylistically, languages and challenge you, the singer, to find your authenticity through the musicality and style and character of a piece. They do NOT coach technique. (Unless you have a voice teacher/coach in ONE person). They are well-versed in the repertoire, can suggest repertoire, and can make OBSERVATIONS about technique that are perhaps not being accessed fully in regard to certain stylistic things. Great coaches will say, for example, "You aren't accessing that vowel in a way that allows you the legato you need in that phrase - talk to your teacher about that". They will not tell you what you are doing wrong technically, nor what to do to fix it. If they do - as many pedagogues will say - please run. This is a voice coach who has stepped over the line of demarcation! Even if the coach is a singer him/herself, if they are not hired to TEACH you, they have no business talking to you about the technical behavior of your instrument. They are their to coach the material.
Great coaches know this distinction and thus, can work from THEIR expertise - which is so necessary for singers.
Simply put - teachers work on the voice as instrument; coaches work on the repertoire; and all that entails!
Opera culture recognizes this "team" vehemently and opera singers have their teacher and coaches in place - working with several coaches who specialize in different languages, repertoires, styles and the list goes on.
Music Theatre singers don't always create this team as clearly, and I really need to encourage you to!!! Sadly, MT singers tend to see coaches more and teachers less - and often do their instrument a huge disservice by working backward: they work with coaches on the music and not on the building of their instrument. They see a teacher when they run into vocal problems (often at the recommendation of a voice coach who knows something is wrong!) and often do not realize that the voice building and instrument need knowledge FIRST!
Guess what? wrong order. Get thee a VOICE TEACHER and have your coach for your musical development, repertoire searches, and your dramatic coach for your presentational development. Your coaches can often recommend teachers they work with or know of, through the singers they coach.
Your team is very important in your development, AND in your maintenance as a singer and performer and as an artist.
Just recognize the distinction and hold fast to that.
You may have a teacher who does work with your music too - especially if they play piano. You may have a teacher or coach who plays as well as gives you dramatic work and language work.
Even though some of us can do more than one discipline, I certainly recommend you still have a pianist/coach you work with if nothing else than another set of ears for musical things, new repertoire ideas and the like.
For opera and MT alike - get involved in workshops and masterclasses that challenge you and motivate you further in your development.
Ask questions; explore! seek!!!
Know what you are looking for and know the differences as you explore teachers and coaches.