The conflict you’re seeing is because both (ki)Honchoushi [本調子] and Niagari [二上がり]are relative tunings. The as yet discussed sansagari [三下がり] is as well. As long as your strings match one of these relationships, you’re in tune.
More specific, set tunings have additional names pulled from sangen [三弦], shakuhachi [尺八] or genre specific systems. Which I’ll discuss in brief now.
Let’s say we want to tune in the key of A (@442hz); perhaps our singer has a particularly low voice or the nagauta/jiuta piece we’re playing requires it.
We first tune the the ichi-no-ito (the thickest string) to our desired key. After getting that first string in tune, we move to the second string.
For Honchoushi, the string is tuned to an equivalent fourth to the first string. So if your first string is keyed to A, that means your second string must be keyed to D. Your third string is then keyed to the same key as your first. Your final result would be A - D - A.
This tuning could be referrred to as ipponhonchoushi [一本本調子], ni shaku ni sun [二尺三寸], or mizuchoushi 水調子, depending on the naming system.
If, instead we needed Niagari, that second string is raised one more step from honchoushi. In our base example, this would make your tuning A - E - D.
In Sansagari, the third string is lowered one step from honchoushi’s original tuning. Meaning we end with A - D - G.
In my experience, my teachers usually tune by ear or with a tuning pipe (調子笛). When we use electric tuners, the standard has been 442hz, but that could be a regional thing.
See the attached image and website for more references.