$beta is an undocumented identifier, that would apparently be used by beta testers.
$opnick/$nopnick/etc. were added in v5.0 and were later replaced by $nick() in v5.60.
They are depreciated identifiers, except $opnick is still used in On Op/Deop, etc
$nickmode appears to be an undocumented sibling to $prefix that doesn't appear in the versions_all.txt
I could track down exactly when $nickmode first appeared, if you really need to know.
PS. Before asking about any undocumented $identifier or /command, check to see if it's just an old and depreciated command, in this complete versions.txt
. Many times, a command is taken out of the help file, or never even makes it in... but will usually appear in an a versions file.