So in the Recaptcha v3 config page you can pick the request method between POST, CURL, and SOCKET.
Which one should I pick? I don't understand the differences. Is one more secure or better than the other? I will probably try each one for a few days or something but that still doesn't help me understand what is what.
No, it's not really a "better than" situation, and is more just "what is actually available / allowed." The selections there already attempt to determine which options are actually available on the server you're using phpBB on, and will disable the selections you cannot actually utilize.
Maybe it should have just be left "100% automatic", and select whatever is determined to be available without bothering to make it "a choice." But there is a choice for what interface to use when invoking the reCAPTCHA library, so there is this corresponding configuration selection in the reCAPTCHA plug-in configuration UI.
This current situation is the improvement and resolution of how -- although technically the older reCAPTCHA library also gave you a choice of methods -- there was no UI nor any automatic detection about what methods were and were not available. When your phpBB server didn't support the CURL method that reCAPTCHA's library defaulted to, reCAPTCHA simply didn't work for you until you manually edited some files to work around it.
Now you have the ability to confirm which method reCAPTCHA is using, if you happened to know that one or more of the methods had been disabled on your server, for whatever reason.
Not that "reCAPTCHA, specifically" would have been disabled by your server, but "the ability for a PHP application to invoke CURL" would have been disabled for security or other reasons, or "the ability for a PHP application to invoke the sockets interface is disabled on this server", etc. Which would have affected both reCAPTCHA and non-reCAPTCHA uses alike.