Persistent= directive. Quoting from the manual:
Takes a boolean argument. If true, the time when the service unit
was last triggered is stored on disk. When the timer is activated,
the service unit is triggered immediately if it would have been
triggered at least once during the time when the timer was
inactive. This is useful to catch up on missed runs of the service
when the machine was off. Note that this setting only has an effect
on timers configured with OnCalendar=. Defaults to false.
Since the desired behavior is kind of hard to describe, this is probably the best keyword that could have been selected.
Another option, not using systemd could be using fcron, which is an enhanced version of the traditional cron now implemented as vixie cron on GNU/Linux systems and others. You can use it's
bootrun option in your tables. See an example here, quoting: