Exactly. The word just means ‘artisan’ or ‘craftsman’, with no exact specification of what one specializes in. If you wanted to say ‘woodworker’, you’ll have to specify it: ‘a wood-working tektōn’.
We get the idea of Joseph and Jesus being woodworkers from an early belief that Jesus made plows and yokes, and it just kind of grew from there. (There’s a symbolic meaning in this: the yoke evokes the wood of the cross. The horizontal beam of the cross is sometimes called patibulum, which means ‘yoke’.)
Now while the idea of Jesus making yokes and other mundane implements is still plausible, the type of carpentry we see in some images - where Jesus and Joseph make large, high-quality furniture - is a bit unlikely, because high-quality wood was rare and were usually imported. In other words, there’s a very slim chance that imported decent lumber (the cedars of Lebanon, etc.) will end up in a small backwater rural village like Nazareth.
As oneofmany implies, at that part of the world you’d get more jobs and money working with stone - which were in abundance compared to wood. With stone, you could build houses or even make bowls or cups or jars. (Most Jews took the purity laws of the Old Testament seriously; unlike pottery or wooden utensils, bowls or cups made of stone are thought to be immune to ritual impurity, which is which they were in high demand, especially for the more pious folks.)
The only thing we know about St. Joseph and Jesus is that they were both artisans. But what exactly they worked on, we don’t know. It’s not like they would have worked only with wood or only with stone; maybe it was both. Maybe they built houses or made stone cups; maybe they made yokes or plows; maybe they also doubled as repairmen, fixing boats or some small furniture or some leaky roof.
In fact, many artisans in those days were not just artisans to the exclusion of other jobs, nor were many farmers just farmers and nothing else. Most people were not stuck to one single kind of job: a lot of working peasants would have had a ‘side job’ of sorts, because after all it helps bring food to the table. Farmers, for instance, also functioned as an occasional artisan or laborer (especially in villages). They farmed for food and they practiced some other trade on the side.
The Galilee was the most naturally fertile agricultural region in all of Palestine, and many people around Jesus would have been subsistence farmers. The grandsons of one of Jesus’ ‘brothers’, Jude, were farmers who owned a small plot of land; it’s not unlikely that many of Jesus’ other relatives were farmers too. It’s not too much of a stretch to imagine Joseph or Jesus doing farm work at some point in their lives in addition to their artisanry.