Weaving was very common in Ulster. Mostly it was a rural activity (because you needed ground to grow flax). Most weaver families had a hand operated loom in their house and wove at home in the winter months when no agricultural work was available. They then sold their produce at local markets. Originally they wove cotton (imported from the USA) but following US independence, and interruptions to supply, they switched to linen because that could be grown in Ireland, whereas it’s too cold and damp for cotton. There are no specific records of weavers. Every agricultural labourer was also a weaver. Mechanised factories gradually made home weaving obsolete and by 1900 it had pretty much disappeared.
Belfast has about 50 Presbyterian churches. Not all were open in 1809. Belfast expanded enormously in the 1800s and many new ones were built. Copies of the records of most Presbyterian churches are held in PRONI, Belfast. Few are on-line and so you would need to either go to PRONI or get someone to do it for you.
Here’s a bit of background about weaving in Ulster:http://www.irish-genealogy-toolkit.com/flax-plant.html
Here’s a link to the church records:http://www.proni.gov.uk/index/research_and_records_held/cata...