How do I handle a mistake in a mathematical proof that does not invalidate the main conclusion of our paper? I recently found a mistake in a mathematical proof of a result I published in 2016. In a nutshell, a subtle error led me to claim that two conditions were equivalent. Unfortunately one of the two implications is not true. Luckily, the main result of our paper relies only on the correct implication so the main conclusion of the paper is correct. How should I handle this?
The most important step to take is to contact the journal and tell them that you wish to publish a corrigendum to your article.
Journals have systems in place for making the existence of a corrigendum clear to readers who search for the original journal article. (So does MathSciNet, by the way—which reviews journal articles, not arXiv articles.)

When people cite articles, they cite the journal versions (when they exist, such as in this case). Therefore people who are reading a future article that cites your journal article will believe, quite reasonably, that they will be accessing the best version of the scientific record if they go to the journal article. A correction on a personal web page or the arXiv does not accomplish this important contribution to the scientific record, causing all kinds of problems (as user124672 mentioned in their answer).

user124672 also makes the valid point that publishing in a venue where somebody disinterested in the article has version-control power over the article (that is, editors of journals) is very important for establishing what is and isn't the definitive version of the article—which is again crucial for the scientific record. A corrigendum is the process by which an author can convince an editor to use their version-control power to update the definitive version in rare circumstances (and the rarity, and necessity for a disinterested party to agree, are features and not bugs of this system). Precisely because authors can update their arXiv and personal web page articles any time they want, such updates cannot ultimately be considered trustworthy in the careful, scientific sense.

By all means, also correct the arXiv version (with a clear explanation of what has changed), as well as the version on your personal web page—but only as supporting actions to the crucial one, which is to correct the scientific record by updating the journal article itself.

(One could imagine a world where later articles cite arXiv papers even when journal articles exist, and where arXiv has some structure in place to prevent frivolous author updates; in that hypothetical world, one could reevaluate the relative importance of the above steps. However, in our actual world, where citations are invariably to stable journal articles, I stand by the above statement.) (One can also reasonably criticize the limited-access and even financially predatory nature of many mathematical journals—I have done so myself—but that discussion about the optimal relationship between academia and capitalism does not alter the facts above.)