To begin with, I shall attempt to define these labels in terms of self-identity (i.e. how deaf people use them).
Deaf (with a big 'D') is used by people whose home is in the Deaf community. They will use sign language as their primary language. It is a cultural label not a medical label, so it isn't defined by how much hearing loss (HL) you have [see below], but by whether you share your life & beliefs with other Deaf people.
However most people with a hearing loss use spoken language (by preference or circumstances) and are not part of the signing Deaf community. Many of this group prefer to use the HOH label as their home is in the hearing world and their self-identity is as a person with nearly-normal-hearing, barely different from other fully-hearing people. Other people with hearing loss prefer to use the deaf label as they see their deafness shaping their identity. It's a matter of personal preference whether to use HOH or deaf so a person with profound HL may see themselves as 'HOH' and a different person with mild HL will describe themselves as 'deaf'. You can even use the word 'HOH' to describe yourself to one group and 'deaf' to another. The term "oral deaf" is sometimes used to distinguish people who communicate orally rather than use sign.
Moving from how Deaf/deaf/HOH people use these labels to how hearing people use them is where is gets complicated! Especially when hearing people often use the label 'deaf' in a negative way ("deaf & dumb"). Plus 'deafness' has legal and medical definitions as part of obtaining welfare/social/medical benefits. So a welfare worker may tell you that you are not deaf because you don't meet their criteria. Note that they are using the labels differently from a person using them to define their self-identity.
Finally there are audiological definitions of HL measured in dB and are classified as mild, moderate, severe & profound. These are functional definitions rather than self-identity labels.
P.S. As far as I know, only drphil uses the term 'DEAF' (all caps)
. I personally feel that "deaf" covers that sufficiently for my use (it may cover a range of "deafness", but at it's end is your word "DEAF") and "Deaf" covers the alternative meaning. I am not saying you can't use that word from now on, actually go ahead as now I'll know your specific meaning. I can understand where it would come in handy. With some people using "deaf" and "hard of hearing" almost interchangeably nowadays, you can use it to make a distinction where "some residual hearing" and "no hearing" makes a difference.