James,
There was no change to the borders between Derry & Tyrone in the period that you are interested in. Derry (which used to be named the County of Coleraine until the early 1600s) had some changes in the very early 1600s but they only affected Donegal and Antrim. Not Tyrone. And were 200 years earlier anyway.
Bear in mind that the spelling of the surname will have varied all the time so keep an open mind on Gilmore/Gilmour/Gilmoore and other possibilities.
Looking at the Muster Rolls c 1630 (published by RJ Hunter) I see about 13 Gilmores in Ulster then. Several in Derry/Londonderry and Donegal. Just one in Tyrone – a James Gilmour – who lived somewhere on the Abercorn estate, in the barony of Strabane. He had a sword and a pike which indicated he was not too badly off. (The number and type of weapons you had generally reflected your economic position). So, you know there was at least one Gilmore family in Tyrone in the 1630s.
Re the other Gilmores in Derry and Antrim, it was a common surname. In the 1901 Irish census there were over 3000 people named Gilmore (including variants) of which 52 were named Samuel. It would have been almost just as common in the 1700s.
Regarding tracing back to Scotland, it’s rare but it can occasionally be done. Usually with families that had a bit of land and money and so left a few records behind (leases, wills and gravestones). Gravestones were not common in the 1600s. Only very wealthy people had them. But I did attend a lecture a year or two back where a Boyd family in North East Antrim who arrived around 1610 had successfully traced their roots to Kilmarnock in Ayrshire and had found a living descendant in London, but from Kilmarnock originally, whose DNA matched the Irish branch.
You ask about DNA. The North of Ireland Family History Society is running a special DNA project focussing on Ulster families. You might want to participate. You don’t need to be an NIFHS member, if you don’t want. You can send them your DNA (they have test kits at a special prices) and contribute your DNA to the project. Martin McDowell their Education Officer runs courses on how to use DNA effectively.
Home - NIFHS.org
You ask about the United Irishmen, and wonder whether your family left because of the uprising. The uprising started on 7th June 1798 and was over a few days later. Your family arrived in Philadelphia in September. A typical Atlantic crossing then was 6 to 8 weeks. So they may have left Ireland in July 1798. They could have bene involved in or affected by the uprising. On the other hand plans for migration often took some months to organise, especially if a property had to be sold, tickets booked etc. So it’s possible they were already planning to leave before the uprising.
You ask about publications on the 1798 rebellion. It gets a few pages in Richard Killeen’s “A short history of modern Ireland,” but there are other more detailed books available (often at a hefty price). It was an attempt to overthrow the Government and was seen as treason (understandably). Martial law was declared and the British authorities responded fairly ruthlessly. The United Irishmen were full of informers and so the authorities knew roughly what would happen and prepared accordingly. The United Irishmen hadn’t many weapons and little military experience so they were beaten fairly easily. In addition, though there was strong support in some parts of Ireland, not everyone did and so that contributed to the defeat too. Many Presbyterians supported the United Irishmen then but soon after the organisation collapsed and ironically by the early and mid-1800s most Presbyterians were happy to be loyal to the Crown again, especially as the economy in the north benefited a little from the Industrial Revolution and a generally improved economy. Today the average Presbyterian in what is now Northern Ireland is extremely loyal to the British Crown. Most don’t see themselves as Irish at all, but British.
The Flax Growers list is a useful tool but it’s not a census. Not every household grew flax and so many folk in Ireland in 1796 won’t be listed. Until the American War of Independence, many weavers had woven cotton which was imported from the Carolinas (it won’t grow in Ireland. Too cold and wet). The British blockaded the Carolina ports and so cotton supplies stopped. Consequently, the weavers needed an alternative source of cotton or a different product. They did use wool but found that linen was more in demand. Linen comes from the flax plant which does grow well in Ireland, and so the authorities encouraged flax growing and gave financial incentives i.e. free spinning wheels. But to be on that flax growers list you needed a few acres of land, so it’s mostly a list of farmers. The average labourer or someone living in a town without any land normally wouldn’t be able to qualify. Do you know what your family’s occupation was in Ireland? That may tell you whether to expect them on the flax grower lists.