Upvote:3
The word that is translated Presbyter in both places (or elder in some translations) is πρεσβύτερος or presbyteros. (Strong's G4245)
There were four types of leader in the early church: 1. apostles (ἀπόστολος or apostolos meaning delegate or messenger) 2. bishops (ἐπισκοπή or episkopē meaning overseer or guardian) 3. presbyters (πρεσβύτερος or presbyteros meaning elder) 4. deacons (διακονέω or diakoneō meaning servant or minister)
The Presbyterian church takes it's name from the presbyterian method of church governance. It's a "bottom up" approach instead of the "top-down", episcopal method favored by the Roman Catholic, Anglican, and Methodist churches. It's a church governed by the presbyters instead of the bishops. It's not always clear from the Biblical texts who had which responsibilities but in the Catholic church, presbyters ended up being called priests. That carries over to the protestant denominations. The presbyter or elder, does what a priest does in Catholicism: lead worship, administer sacraments, council, teach, etc. Churches group together and then groups group together to form a presbytery which is a council of Church elders who govern the churches within their respective groups.
So, I suppose the short answer to your question is that they're both.