Firstly: The term “rite” is fine, but only in its appropriate context. For example the Melkite Church is not the “Melkite Rite”, but rather it is the Melkite Church which uses the Byzantine Rite.
One of the key reasons for this distinction, and for insisting that the Eastern Churches not be called the “Eastern Rites”, is that a Rite doesn’t make for a distinct hierarchy, but a Church does. For example, the Ukrainians and the Melkites have distinct hierarchies, as in they don’t share priests and bishops, and have their own Synods and such, even though they share the same Rite. On the other hand, you can have more than one Rite within a single Church, as is the case with the Latin Church which has a number of Rites but only one hierarchy.
As for why your Latin sources wouldn’t have embraced this terminology, I would imagine it’s just from ignorance or habit in most cases. This is the terminology of the Vatican and of Canon Law, so any other uses are merely mistakes.
Secondly: I’m not sure I understand this question. There’s no threat to unity by speaking of different Rites in the Church; heck, there’s no threat to unity by speaking of different Rites and theologies within just the Latin Church (the Dominican Rite and theology versus Franciscan theology or the Mozarabic Rite, for example). The term Church can be used for the entire Catholic Communion, and for particular Church hierarchies within it, such as the Latin and the Melkite.
Perhaps you’ve misunderstood the objection to the use of the term “Rite” in an inappropriate context to mean a general rejection of the term all together? Rite is a fine term, and is used all the time throughout the Catholic Church, it just has a very specific meaning and should be used appropriately.
Thirdly: I personally don’t know of any official document addressing this, I only know of how official documents refer to the Eastern Churches, and that’s as Churches and not Rites. The term Rite is defined by Canon Law, and can be found here in the
Eastern Code of Canons and the definition of Church is found
here. As you can see from the definitions, they are distinct terms with specific meanings, and not interchangeable.
Hope that helps! Peace and God bless!