While the many spelling versions of Brigid’s vary due to local dialects, they also vary according to Gaelic grammar rules, which change how a word is spelled and pronounced depending on how it features in a sentence. The English language doesn’t do this, so it is an aspect many anglophones miss in trying to understand why the name is spelled so many ways.
This also speaks to the grammar applying in many regions of differing dialects before the Gaelic languages were later standardized. Many of the spellings seen don’t refer to Her name itself, but to what anglophones would know as a combination of words in a sentence, like ‘of Brigid’, or ‘to Brigid,’ which are denoted in Gaelic languages through spelling changes, like adding the ‘h’ after the first letter, or later in the name, or an ‘e’ at the end, rather than writing the article out as we do in English.
A good example of this is my name, Erin, which is the anglisization of Éireann in the Irish, and means ‘of Éire,’ or, ‘of Ireland,’ as in, Bus Éireann, which means, the bus of Ireland, its national bus service. But, to say Ireland itself in Irish, it is Éire. Her name itself is written as Brighid, Brigid, or Bríd in the Irish, and Brìde in the (Scottish) Gaelic, and the other forms like Brìghde, Bhríde, and Brighidh are forms of saying, ‘of Brigid’, as in the Well of Brigid, Tobar Bhríde in Ireland, or Tobar Brìghde in Scotland.
My understanding of the Scottish Brìde is that the Scots had a habit of taking that altered form of the name and then just using it as the name itself, so Brìde is the altered form of the Irish Bríd, with the ‘e’ on the end. The Scots also did this with the Irish Seamas (said, SHAY-mahs), meaning James, which is altered to Sheamais when addressing Seamas by name, like, hello Seamas, which is now anglisized to the common Scottish name, as it is said in english, Hamish, as the ‘sh’ is pronounced as an ‘h’ at the beginning of a word (silent elsewhere though), and the ‘s’ is pronounced as ‘sh’ when sited beside an ‘i’ (and an ‘e’).
My understanding of the ‘h’ in the middle of Brighid’s name is that in the older forms of Irish the name was written with a dot over the g, to denote its silence, but when typing came along and the dot could not be added, it was change to an ‘h’ next to the g, as ‘gh’ in the Irish language is silent. In modern Irish, the name has been standardized to better reflect its pronunciation as Bríd, said “breed” in most dialects, but is said “breedj” in the Ulster dialect of the north, and that is carried over into Scotland as well. This is the general pronunciation of all forms of Her name which don’t have an ‘e’ at the end. Those with the ‘e’ are generally pronounced, “BREED-ah” in most Irish dialects, but “BREEDJ-uh” in the Ulster dialect, and in Scotland. Also generally the emphasis is on the first syllable in Gaelic languages, unless otherwise noted by the marks over the letters (fada in Irish, grave in Gaelic).
The form written Bridie, pronounced BRIDE-ee, is a Broad/Lowland Scots pronunciation of the Gaelic Brìde, which is said, BREEDJ-uh, as noted above.
The standard English spelling of Her name is Brigit, which is pronounced how it is often otherwise spelled, Bridget.
While I have studied both Irish and Gaelic for a number of years, my skill level with either is still not beyond the basic beginner level, but I have gleaned much of this information over those years, and some of it has been provided to me by acquaintances who are linguistics scholars in these languages. I don’t therefore have any proper citations to offer for the veracity of these guidelines, but am confident that they are sound. However, should a linguistics specialist have need to correct me here at all, please don’t hesitate to do so.
Hope this helps in understanding some of the workings of the Gaelic languages.