Carpenter is a tricky term because wood is used in so many ways. This is strictly UK biased:
In building terms we distinguish a first and second fix carpenter. First fix is mostly structural and second fix is the rest. If your hair is wet its probably first fix - roof trusses and the like. Second fix will include things like skirting boards. It's not a hard and fast rule and will probably involve the same people. These trades are known as "chippies" (sing. chippy).
We also have cabinetry (your furniture smith - nice one) and other wood working trades. They are all called carpenters. A chippy is a carpenter and I think this a rare case where the nickname is more common than the real name. Civilians will often call a chippy a "builder". A cabinet maker is never known as a builder or a chippy.
The chippy thing is because a very small piece of wood is a "chip".
I think I know why you are a bit confused about the word carpenter. In the US and elsewhere, houses are mostly wooden. When a house is nearly all wood, then the word carpenter should mean builder. In the UK and elsewhere houses are built of sticks and bricks, mostly bricks.
Depends where your are, chipper is common to some, others say 'fish shop', some 'chippy'. Other people say they're "getting chips", or "having carry-out" and leave the place to be entirely implicit.
That said, I've never seen it called the "fish and chippery", but it wouldn't surprise me to see that used.
In building terms we distinguish a first and second fix carpenter. First fix is mostly structural and second fix is the rest. If your hair is wet its probably first fix - roof trusses and the like. Second fix will include things like skirting boards. It's not a hard and fast rule and will probably involve the same people. These trades are known as "chippies" (sing. chippy).
We also have cabinetry (your furniture smith - nice one) and other wood working trades. They are all called carpenters. A chippy is a carpenter and I think this a rare case where the nickname is more common than the real name. Civilians will often call a chippy a "builder". A cabinet maker is never known as a builder or a chippy.
The chippy thing is because a very small piece of wood is a "chip".
I think I know why you are a bit confused about the word carpenter. In the US and elsewhere, houses are mostly wooden. When a house is nearly all wood, then the word carpenter should mean builder. In the UK and elsewhere houses are built of sticks and bricks, mostly bricks.