“open source” (capitalized or not) means that Open Source initiative agrees with the license. There are some conditions that I don’t remember.

It’s different from “source available” - open source should let you fork and reuse it.

For example there is microsoft shared code license, that’s “here is the code, but copyright is still ours, you can’t do anything with it but look”.

I have no idea what kind of license (if any) is unreal engine.

That's a misleading, obfuscating way to make the difference. I guess the OP means an OSI-approved licence.

If you write your own licence (not recommended, but some developers and especially corporations do) it could be even fully compliant, but not approved.

"open source" and "free software" are two words for the exact same thing.

Both of them are pretty poor descriptors. "open source" doesn't convey the legal freedom you are granted (as you have just found out), and "free software" makes it sound like it's just about price.

If someone lets you see source code but doesn't allow you to do anything with that code it's not what people would call "open source", you could probably call it source-available or something. "open source" has a specific legal definition that means code released with a permissive license.

It's under EULA.

There is a trademark for "Open Source", and if you want to use it without written approval from OSI, you have to use it for something under an OSI-approved license: https://opensource.org/trademark-guidelines#Usage_That_Does_...

I think this is the closest you can come to a "formal definition", short of a law defining the term.