The NBER is a private think tank. You may be confusing them with the Bureau of Economic Analysis, which is an independent federal agency.

No, it is the NBER. Even the BEA says the NBER officially defines recessions:

https://www.bea.gov/help/glossary/recession

There’s no official definition. It’s like saying you’re sick, you can be sick even if doctors can’t point to what’s wrong with you, and you aren’t necessarily sick even if you have a condition. The two quarters thing is just a really widely recognized rule of thumb. The idea that there’s some sort of official definition every economist agrees on is incorrect. The idea that there’s an organization that owns the word recession is also wrong.