This is exactly what I believe is a motivation for attempting to "make uniform" a set of terms. It's like trying to identify a family of siblings or floors of a building. I'm not saying whether or not it makes sense to add more terminology (in this case abbreviations) to try to emphasize such relationships. I personally have no problem using the distinct terms without the added "Service Provider" family to tie them together. I do strongly believe though that we could greatly benefit from a "basics" topic/page that clearly names and outlines these "layers" of website-operation to bring better awareness of their existence and how they interact with one another and the certificate-acquisition process. This might make the public a little wiser and better organize the approach we all take collectively when aiding people. I'm a big fan of having well-constructed, specific, standardized knowledge points to use/give as reference.
For example: