Colleen,
Most Irish censuses pre 1901 have been destroyed but some fragments exist for various reasons. It appears that the 1831 census for Co Londonderry was out of the Public Record Office in 1922 for research purposes and so survived the explosion and fire. Some other fragments of pre 1901 censuses have survived for various reasons too. For example, when the old age pension was introduced in 1909, you had to prove you were 70 or older. The 1841 & 1851 censuses were used for age purposes, to check whether you qualified. Knowing you were, say 3 in 1841 would demonstrate that in 1909 you were 71, and so eligible. The clerk doing the verification in the PRO often transcribed the relevant entry with all members of the family and their ages, and if that happened that information has survived. But many people had baptismal certificates or other proof of age, and so not everyone needed a census extract. So the extracts are patchy.
Yes Eugene is the anglicised version of Eoghan, which as you surmise, is pronounced Owen. (And can be spelled Owen too).
If the family were laborers then that’s not the same as a farmer. Laborers might have a little land (usually a few perches) to grow potatoes or flax but a farmer would have anything from 2 to 50 acres. More importantly a farmer would normally stay in the one place. You don’t spend time improving your farm for a few years and then move to another location if you can avoid it. Farmers would also be listed in the tithe applotment records (which were essentially a tax on land to pay for the Church of Ireland). No point searching them if the family were laborers. And laborers often moved around to follow the available work, so they can be hard to locate in official records.
Most laborers in Ulster were also weavers. In rural areas most weaving was done at home using hand loom weaving machines, such as are still used in the Outer Hebrides in Scotland to make Harris Tweed. These machines were collapsible, so as to be stored out of the way when not needed and for ease of transport should the weaver move home, as they often did.
Most weavers in Ulster were labourers who earned a bit of extra money by weaving in the winter months when there wasn’t much labouring work required on farms. This meant that labourers in Ulster had a slightly better standard of living than elsewhere in Ireland. It also gave them some ready cash (in a society that mostly operated by barter) for the things that could not be bought by barter, eg a ticket to America. At one time weavers wove a lot of cotton but the interruption of supplies from the southern US states during the American War of Independence in the 1770s meant they focused on other materials, notably flax (which linen is made from) and which grows well in Ireland in contrast to cotton which won’t grow at all. They did weave other products eg calico and wool as required, but by the 1800s it was mainly linen. Linen made at home was taken to the local linen market and sold there.
As the 19th century progressed, water powered linen mills were introduced all over Ulster. These were faster than home weavers and often made better quality material and so gradually made the home weaver redundant. In addition, the factories also mostly employed women and children (being nimbler and cheaper) so this impacted on male employment too. At a time when farms were starting mechanise and so needed fewer labourers, the average labourer/weaver therefore faced a bleak future and so these combined changes were a major factor in many a labourer’s decision to leave Ireland during the 1800s.
A little more information on this link: https://www.irish-genealogy-toolkit.com/flax-plant.html
Good luck with your research in Co Derry. If needing somewhere to stay, I can recommend Laurel Villa in Magherafelt. Owned by Gerardine and Eugene Kielt who is a local historian.
https://discovernorthernireland.com/accommodation/laurel-villa-townhouse-p678451
