Learn Spanish (Latin America): El piloto está por soltar el amarre. - The pilot is about to release the mooring.

While va a soltar just means "is going to release" (which could happen in five seconds or five days), está por soltar emphasizes that the action is highly imminent. It translates best to "is about to release" or "is on the verge of releasing."

I've also learned that estar a punto de means "about to". Can I use that here instead?

Yes, you can! In Latin America, estar por + infinitive and estar a punto de + infinitive are practically interchangeable for imminent events. El piloto está a punto de soltar el amarre means the exact same thing.

Doesn't estar por + infinitive sometimes mean an action is "yet to be done"?

It does, but mostly in Spain! In European Spanish, está por soltar would usually mean the mooring "is yet to be released" (a task that hasn't been done yet). However, in Latin American Spanish, this structure is the standard, everyday way to say an action is "about to" happen.

What exactly is an amarre?

It comes from the verb amarrar, which means to tie, fasten, or moor. In a nautical or aviation context, el amarre refers to the physical mooring, tie-down, or rope that keeps a boat, plane, or balloon secured to the ground or dock.