Irnimer,
You ask how common it was to change denomination. I don’t have any statistics but it did happen. Sometimes through a decision to swop for personal religious reasons, but sometimes more through marriage. Tradition was to marry in the bride’s church after which she’d attend her husband’s. But that didn’t always happen and so sometimes if the bride felt stronger about her denomination than her husband, then they stuck with hers. (And I know a man today who attends both his and his wife’s church. So going to both was an option too).
MacLysaght’s Surnames of Ireland says Michael was an English name which in many cases over the years became Mitchell. There were also Scots called Michael but he says they were mostly MacMichael.
Co Londonderry/Derry was heavily settled by significant numbers of both Scots and English, but if we go with MacLysaght and assume Michael is an English name then your family were probably originally Church of Ireland (ie Anglican/Episcoplian). Presbyterian points to Scots origins. Following that idea along, then over the years one branch of your family switched to Presbyterian, perhaps through marriage or perhaps through simple religious inclination. You may never know.
Which church did they go to? Well the Church of Ireland (and the Catholic church) uses the parish system and so you normally attended a church in the parish where you lived. (Some parishes had more than one church in them). Bovedy is in Tamlaght O’Crilly parish. There’s an upper and a lower church. I don't know the area well enough to advise which your family attended. Gravestones might be a clue though if you know where they are buried.
Presbyterians in Ireland didn’t use the parish system and which church you attended was simply based on personal preference. Not always the nearest to where you lived. The 2 nearest to Bovedy were Bovedy (records start in 1841) and Churchtown (records start 1840). There’s a copy of both Presbyterian churches records in PRONI. I don’t think they are on-line anywhere.
However I have a clue for you. I spotted a marriage for Eliza Michael of Bovedy on 16.1.1862. She was daughter of William Michael, farmer, and she married John Patterson in Kilrea 2nd Presbyterian church. So there’s a strong hint that the Presbyterian side of your family may in fact have used that church. So check that church’s baptisms too. (Copy in PRONI).
There’s 59 Michael marriages in the area (Ballymoney registration district) between 1845 and 1900. You could go through them all and probably get quite a bit that way. For example, I see Elizabeth Michael aged 21 marrying Thomas Churchwood alias Kirkwood (also of Bovedy) on 2nd June 1846 in Tamlaght O’Crilly Church of Ireland.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
Tamlaght O’Crilly’s early records for both churches were lost in the 1922 fire and they now have nothing before 1895 for the upper and 1881 for the lower, save for non RC marriages which start in April 1845 and which are on the irishgenealogy site. (From 1845 onwards, non RC marriages were copied to the civil registrar, so though the church’s own copies have been burned, the Registrar’s have survived).
So you can get back to 1864 with the statutory birth and death records and to 1845 with marriages. For the Presbyterians you may be able to get back to 184/41 with baptisms but with the Church of Ireland you won’t get back earlier than the statutory birth records.
You ask which denomination would be closer to Methodism-Episcopalian. The answer without question is Church of Ireland. The Church of Ireland is an Episcopalian church. Methodism was a breakaway sect from the Church of England/Ireland (C of I and C of E are the same denomination). So it’s common to find Methodists who used to be Episcopalian. Much rarer to find Presbyterians with Episcopalian inclinations.
Glad you found the advice about checking for wills in Dublin helpful. I suspect comparatively few people know that which is why I specially mentioned it in that reply.