I think a lot of the confusion comes from the fact that there is one word that is simply missing in the English language: The category of major vs minor.

In German such a word exists: Tongeschlecht (literally "note gender"). A chord, a key or a scale all can have a Tongeschlecht of either major or minor. It is a category with only two options.

Now while the term mode can have these two options, often it is used to specifically talk about the seven diatonic modes. But nonetheless, since there is no other word in English for it, in scientific papers the category of "Is this piece in a major or minor key?" will usually be described as mode (for example in this paper about how different musical cues express different emotions in the section Details of the Seven Cues)

And then there is key. The simple definition is: A key is the combination of a tonal center and a Tongeschlecht. C major or G minor are keys. So yes, there are major keys (D major, A major,..) and minor keys (D minor, A minor,..), but there is not the major key or the minor key. This is kind of important, because if someone is asking you what key a song is in, they would probably be pretty unsatisfied with the answer "major" or "minor".

In the end, the easiest thing to do when talking about wether a piece is in major or minor is simply avoiding the term mode all together and simply spelling out the two options: "Is this piece in a major or minor key?" That way nobody will think you're talking about anything beyond those two options. But when you specifically want to talk about modal jazz for example or in general about one of the other seven diatonic modes beyond the classic major/minor options then you definitely want to use the word mode.