log on has the sense nowadays using a sign in procedure to enter an internet or electronic space.
log in has historically meant to sign in, as one might sign one's name on a duty roster or log that shows who is present at work or a meeting.
You're welcome. Another aspect of the older use of log in is that the writing of one's name on the log would be accompanied by a notation of the time and date of logging in, and logging out. In this case the word log is related to a log (a daily record, or record of a shift of work) created over time, as those kept daily by the captain of a ship while on a voyage.
With computers, "log on" refers to getting online, "log in" to entering a particular software environment. "In" usually refers to entering a smaller "space" than "on", so we might log "in" to a website but "on" to the internet, log "on" to MVS but log "in" to a CICS region.
An application may run "in" Windows, but Windows runs "on" my PC.
In 1970's when working at IBM as a mainframe technician I used to log on my work station to the system over a coax cable.
Later on between 1973 and 1975 a few guys Robert Metcalfe being one of them took out patents on the Ethernet - he later left and started 3COM.
DEC (Digital Equipment Corporation - now defunct-dead) installed it using huge cables. Over time the cable changed to cat5 and the famous RJ-45 connectors were used as we know today.
When servers were used over the Ethernet LAN the term login seemed to have been born.
Why? I have no idea.
PS-Xerox engineers also invented the mouse and what became Apples OS - Steve Jobs knew the Xerox CEO and one day the CEO of Xerox showed Steve what his engineers were doing to waste time and money. Steve asked if he could have it and the CEO (pronounced - idiot) gave it to him. That is why Steve couldn't patent the mouse.
I once worked for Xerox 1in 1965 and their equipment was truly amazing.