Deb,
Logan is a fairly common name in Ireland, especially in Ulster. (In the 1901 census there were 2168 in Ireland). It’s unlikely they are all related though some will be obviously. The problem that you (and others researching the 1700s in Ireland) are up against is that we just don't have any comprehensive records. Apart from a few major landholders, and the odd family with knowledge of their specific origins, we do not know who came over in the 1600s or specifically where they came from, though we know in broad terms which counties they originated.(The occasional Irish gravestone mentions a place of origin in Scotland. That's very rare but I have encountered it a few times). Once the Scots got to Ireland record keeping wasn't great either, and we didn't compile any lists when they starting leaving again in the 1700s. And of course some may only have lived in Ireland for a generation or two, perhaps just 50 years, further reducing the chances of finding records.
Presbyterianism was founded in Scotland in the mid 1500s and brought to Ireland (and elsewhere) by Scots settlers. It was heavily influenced by John Calvin (1509 – 1564), a French theologian, who in turn had been heavily influenced by Martin Luther. Calvin was born Catholic but developed views that were incompatible with that denomination’s beliefs. For example, divine pre-destination. He also strongly disapproved of the arts, save for music, but even that could not involve musical instruments. He sent out pastors to the rest of Europe and his beliefs led to the foundation of not only the Presbyterian church in Scotland but also the Dutch Reformed church and the Puritans in England. So these denominations all have a common origin.
If you were to attend a Reformed Presbyterian church service in Scotland or Ireland you would notice that there is no organ, piano or other musical instruments, and the only music is psalms (no hymns). A precentor stands in front of the congregation and leads the singing. No female Ministers and no woman may sit on the Kirk Session. They tend to favour a strict interpretation of the gospels. If something is not expressly allowed in the bible, then they generally won’t allow it. There is no episcopacy in the Presbyterian church. So no Bishops, Archbishops and Cardinals etc. No hierarchy to control the congregations (though the various Ministers do meet for obvious administrative purposes). No icons or religious imagery, and generally a plain building. Communion (mass) is a special occasion and only held two or three times a year.
Quakers however have completely different beliefs (and don't even have a Minister to officiate), and apart from obviously not following the doctrines of the Roman Catholic church would not normally be seen as having very much in common with Presbyterianism. If your ancestors switched from Presbyterianism to Quaker beliefs, that sounds like a sharp right turn. Presumably they had decided to try something new.