mathematicaDefinitionOfArcCoth comes from ArcCoth, Properties & Relations, see the final sentence and then back up two sentences.

A music-software company gives me free products when I beta-test for them, so I have to give my bug-report-submitting time to the other company.

You can expect the numerical algorithms to give inconsistent results.

Mathematica gives the same false result even if you omit the "N"; the issue here is not a matter of any numerical algorithm.

The issue is rather a lack of coordination and communication between the developer who implemented ArcCoth and the person who authored the "Background & Context" section in the docs. As far as I have found, a fix would require no change to the implementation of ArcCoth; a fix would require only a correction to the documentation. I foresee no fix ever happening, though, because the consensus among the community is that wrong is better when wrong is what Mr. Wolfram chooses.

Thank you. If "Back & Context" said what you said, there would be no issue. Instead, "Back & Context" shows a different expression, one unequal to ArcCoth. Along the discontinuity, the discrepancy makes all the difference in the world.