This is optional, in case you prefer your nick to be shown as something other than <matrix_nick>[m] on IRC, e.g. just <matrix_nick> (without the [m] suffix) or <libera_nick>.
Open a private chat with @appservice:libera.chat:
!nick <new_libera_nick>
Note: This does not change your Matrix nick!
Open private chat with @NickServ:libera.chat, register your current nick on libera.chat (if applicable), and login:
Note: If you already have a registered nick on libera.chat, you can use it at this point to log in. You may want to add the nick you're using right now to your existing libera.chat account as an alias using the step below.
REGISTER <libera_password> <libera_email>
# now wait for NickServ to reply, then proceed to your mailbox for the registration code (paste from email)
VERIFY REGISTER <libera_nick> <registration_code>
# you should be getting logged in now.
# In order to log in next time send the following command, or store your password as described below to automate this step:
IDENTIFY <libera_nick> <libera_password>
To add an alias nick to your libera.chat account, change your nick to your alias as shown above and then command @NickServ:libera.chat to:
GROUP
In order to be logged back in automatically, open up the private chat with @appservice:libera.chat again, and do the following to save your username and password:
!username <libera_nick>
!storepass <libera_password>
Updated. Thank you @Mikaela and @decathorpe!