I think you’re overthinking it. If you’re only triggering updates based on changing “to on”, you can just use the {{ now() }} to record the time of the last trigger.
So I think the below code should do what you want:
template:
- trigger:
- platform: state
entity_id: binary_sensor.bewegungsmelder_schreibtisch
to: 'off'
sensor:
- name: "Bla bla bla"
state: "{{ now() }}"
device_class: timestamp