Can't resist ... am a 'fan' of git ... updates or upgrades.
Since you already have a 'method' ... although 'old' and probably prone to human error regardless of how many times one might have performed them in the past ... suggestion ... maybe actually a suggestion to become familiar with admin of a moodle via git ...
If you have space and ability ... semi clone your production to a clone. Dump DB (sql) and import into new DB name ... same credentials - new DB name moodlegit. Archive moodledata and restore to same location but different name ... like /var/www/moodledata achived ... cp to /var/www/moodledatagit
In another domain or subdirectory of server, install a 3.8.1 code base via git.
in that git acquired directory of code, copy your production sites config.php to moodlegit ... edit config.php and make changes needed to reflect the clone/copy.
Then use moodlegit/admin/cli/ and run the upgrade.php script there. Your DB thinks the cloned site is a 3.8+ ... code is 3.8.1 ... which should upgrade the DB.
If, if that's too much ... just start from scratch ... new git acquired code ... but make it a 3.7.highest where you can then practice the 3.7.highest to 3.8.highest upgrade without fear!
Unlike Howard, I do get lazy from time to time and use the Admin GUI to update plugins/addons. But I do a combo ... have a script in code to 'unlock' (change permissions on key directories), run the GUI updates, once completed I have a 'lockdown' script that changes permissions back to more secure.
Encouragement: once you get a handle on git install/updates and upgrades, you'll wonder why you didn't do git a long time ago. Literally a matter of minutes once you figure out an 'up' script and an 'update' script.
'SoS', Ken