Does a drink made by a juicer have a specific name in English, or is it just fresh juice?

No it doesn’t. Juice is juice. Sometimes we say ‘fresh squeezed orange juice’ though as opposed to ‘juice by concentrate’. But often we don’t clarify.

Nope, just juice. It might be called "freshly squeezed juice" or "fresh-squeezed juice" on restaurant menus and the like.

"Fresh squeezed" juice :)

Straight-up juice might have a signifier added to what kind of juice it is (“lemonade juice,” “orange juice”). When sugar or seasoning is added, then it gets its own name (“lemonade,” “grenadine”).

This is weirdly phrased because it implies there is some way to make juice without juicing the fruit. Like if it comes from a juicer then it's juice I mean those are the coordinate terms. But it sounds like you're trying to distinguish between fresh juice that was hand squeezed versus buying juice already in juice form. Then yeah it would be like "Do you want some juice? I made/squeezed it myself" or "do you want some freshly made/squeezed juice?"

"Fresh Squeezed" is as close as it gets

The default name of a fruit-based drink is juice. We even sometimes call mostly fake fruit-flavored drinks "juice", but it won't say so on the bottle.

It's just \[fruit in question\] juice. In this there may be significant differences between american and british english though, I'm not sure.

Are you by any chance italian? We've got the word "spremuta" for that.

All juice is made by a juicer. It's just that sometimes you are the juicer.

Freshly squeezed [fruit] juice e.g. Freshly squeezed orange juice.

Just a quick note, if a carton or bottle says "<fruit> Juice" and nothing else, it contains NOTHING BUT juice from the <fruit>. No additives. If the carton or bottle says "<fruit> Juice Drink" it will have added sugars, preservatives and other stuff.

Freshly squeezed.

Not really. Fruit put into a juicer is called juice. You *can* make an exception to be more specific and say "[fruit name] juice" but that is usually not necessary.

I’d call it delicious 😉 but really it would be fresh juice. If you want to be specific, you could put the exact fruit(s) after the word fresh, ie. fresh apple juice. Cold-pressed juice is something you might also see, which refers to a certain method of making fresh juice.