I am trying to buy this kind of fish hook (picture below), but I have no idea what is called. Does anyone know what to search for?
The main characteristic of this hook is that the hook eye is solid as opposed to a ring.
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That is very simply called an eyeless hook or a spade hook.
Spade hooks are old school, but there are enough people that still say they're preferable to eyed hooks, in fly fishing at least, you tie both eyed and spade hooks basically the same. Some people pass the line almost arbitrarily through the eye on an eyed hook, but it's easier to tie them without going through the eye.
OK... The real name for that kind of fishing hook is French Snell. Gamakatsu calls it "French Hook Snell", and Mustad calls it "French Snelled Hook". And the specific one you showed here is an "Off-centered French Snell Hook".
You need to wrap around the line instead of inserting it in an eyelet. It's good for very small fish only, and I wouldn't trust it for pikes, walleyes, bass, etc...