I believe "temple priests" is a general term for any priest that belongs to one of the O.T. temples, i.e. Solomon's Temple and Zerubbabel's Temple (possibly Herod's Temple, too?). I don't know offhand if Tabernacle Priests are considered "temple priests".
Melchizedek didn't belong to any of the temples, nor the Tabernacle, so I don't think he is considered a temple priest. Faithful Priest and Priests of Christ are both N.T. and I don't think any N.T. characters belonged to the temples so I would assume they aren't either.
There is not currently a definition for temple priests in the REG so I'm not 100% on this.