Just to add, since I didn't speak about the differences between (1) and (2) or (3) and (4):
You have to use (4) if (and only if) you're going to use a resolvconf(8) implementation other than systemd-resolvconf. For example, I use dhcpcd; for kinda silly reasons dhcpcd and systemd-resolvconf don't work together, so I'm stuck using openresolv as my resolvconf(8) implementation. So I have to use (4). If you're fine using systemd-resolvconf, you should use one of the other 3 modes.
I'm honestly not sure why you would ever use (2) instead of (1). It's like (1), except search paths won't work right for programs that don't use the system APIs. (Maybe it existed before (1) did, and still exists for historical compatibility?)
You have to use (4) if (and only if) you're going to use a resolvconf(8) implementation other than systemd-resolvconf. For example, I use dhcpcd; for kinda silly reasons dhcpcd and systemd-resolvconf don't work together, so I'm stuck using openresolv as my resolvconf(8) implementation. So I have to use (4). If you're fine using systemd-resolvconf, you should use one of the other 3 modes.
I'm honestly not sure why you would ever use (2) instead of (1). It's like (1), except search paths won't work right for programs that don't use the system APIs. (Maybe it existed before (1) did, and still exists for historical compatibility?)