A shastri knows certain texts at the intellectual level, and has been qualified/authorized with adhikar to explain them to others. S/he may or may not know them experientially. I would say that outsiders are typically shastris. Most Indologists are certified as shastra experts and dont necessarily even respect the teachings experientally.
A rishi knows experientally; this is embodied knowing. This is a higher level than mentally knowing shastras. In fact, the rishi's uttering later gets compiled into shastras which other shastris study and try to interpret.
A pandit/purohit has adhikar to perform rituals as per aagamas. May or may not know shastras or have embodied experience. But there are strict requirements to get this adhikar.
There are pre-requisites in the tradition for being given specific kinds of adhikar. I invite people here to post from various traditions and explain the process for obtaining adhikar of specific kinds.
Western "authority" is based on a system of higher education certification that has its own history and politics. We have discussed it many times. Indians have largely adopted this criteria for authority. But such authority is not the same as adhikar in the tradition.