That is all good thinking.
Complete extended harmony chords are often impractical to voice on guitar. For that matter they are frequently not even voiced as theoretically idealized in any music. Their primary function is to have a unique identifying name for a collection of pitches that occur together. And because some jazz & classical pieces have those extended notes beyond the basic triad, this became the standard naming convention.
As you mention, some notes are left out as a matter of stadard practice. And it's at the players, or composers, discretion or intention to decide which ones are important to have or leave out based on the music context.
I've seen situations in which the 7th is omitted. But the 9th or 11th is included along with the 3rd & 13th. So the best name for it is still a major 13th chord. But as you say, if it's only the 13th being added, then it's more precise to just say it's an added 6th, or added 13th chord.
Hope that helps!
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