The code for the trigger can be updated when a new version is published.
All the parts of making a "trigger" work consists of:
1) set-up - type of trigger - What event causes it to run - configuration settings eg. how often
2) the event - What is the Google server detecting to call the function name
3) the code associated with a function name
4) The function name
The time-based trigger needs to be installed. That's the set-up, which consists of things like how often to run it. The event is what is being "listened for." What often happens is, that the use of the word "trigger" gets associated with just one, or some of those parts. And the word "trigger" gets used interchangeably to mean the function, or the set-up, or the installation.
So, that leads to confusion and misunderstandings when talking about triggers.
If you change the code in the function that is linked to the trigger settings, then the trigger will run different code.
If that different code installs a trigger, then a trigger will be installed. The code is what installs or doesn't install the trigger.
Just the act of publishing doesn't necessarily do anything. I just didn't what you to think that just the act of publishing a new version would necessarily do anything. It might, it might not, it depends on what the code does.