The IT Security definition of a "backdoor" is: something that provides open access to the data without the knowledge or control of the owners of the data - who are typically the users.

There's nothing about the legality or not of it or the company that makes the software being aware of it, which is why sometimes you get news about how a software maker having bing discovered to have a "backdoor" in their software and many of the ways the Chinese Government forces companies to provide access to user data, whilst being 100% legal (just like the US) are described as "backdoors".

From the point of view of IT Security specialists a technique having been endorsed by members-of-parliament/senators/congressmen/governments/presidents/monarchs/whatever or not is relevant for the naming of that technique - if it provides open access by a 3rd party to user data without user knowledge or control it's a "backdoor" and using it is "backdoor access".

So it's funny (sad funny rather than "ha ha" funny) how in (mainly American) newsmedia stuff which is 100% legal in China is described as a "backdoor" but the exact same techniques when 100% legal in the US are not describe as "backdoors" whilst technically being exactly that: honest and unbiased news would deem both backdoors or not depending on their characteristics (i.e. are they means of open access to user data without the knowledge of the owners of the data). It's clear the technical term is being misused due to it's association in the minds of people who aren't domain experts with "bad thing".

Normal warrants issued by a normal Court usually aren't considered "backdoor access" not because of their legality but because they're limited and executed by the people inside the company that received the warrants in a case-by-case basis (i.e. they fail the "open access" criteria), but the kind of warrants issue under FISA definitelly was open and forced the companies to provide open access: that's exactly the problem and that along with the absence of Probable Cause is why many consider it to go outside Rule Of Law.

It's unclear if FISA warrants have been used or not to force companies to provide what are (per the technical definition) "backdoors" in actual software implementations, but as we know thanks to Snowden they certainly did force some companies to provide NSA with free realtime access to their systems, and having a NSA server getting copies of any user communications passing through a mobile phone provider is technically "backdoor access" to their systems.

In summary, Engineering doesn't care about politics when naming technuques and beware that legality isn't the same as morality: all the shit that China does is just as as legal as all the shit the US does - after all, the people who make the laws are the one who authorized it.

Personally that was exactly the scary part in the Snowden revelations: the US plus a bunch of other supposedly democratic nations where doing exactly what dictatorships did, by changing the Law to make it legal and then deploying intrusive society-wide surveillance.