FWIW, I’ve always heard and used “closed harmony,” but it seems “close harmony” is the more popular term.
Another quintessential example of close[d] harmony is George Shearing.
One other point. Harmony, when referring to multiple voices, are those parts that support and compliment the melody.
I recall when I used to manage a band and do their live sound, our agents turned up one night to hear them. After they finished their set one of the suits came up to me and said, “That was a really great sound you were getting. The rhythm section are as tight as a fish’s asshole.” Hope that helps.