Mental health and physical health both influence emotional health. Mental health, however, is related to the physical state of the brain. Consider that after a stroke damages part of the brain, a person loses abilities like sight, smell, hearing, balance, memory, reading, etc. Losing the ability to read can bring depression (poor emotional health). Having a broken arm and missing a wedding can bring depression (emotional health). I have several close relatives who have mental health issues. Stroke, blows to the head, and other things damaged parts of their brain and they have lost abilities just like a broken arm causes a person to lose abilities. It is heart breaking at times to know that a broken arm heals, but we don't yet know how to heal parts of the brain when it is damaged.
A person can choose to define depression as mental health. Having seen relatives with damaged brains I choose to separate mental and emotional health. I am fortunate that one of my relatives with a damaged brain has good emotional health, i.e., they are happy and understand what they have.