I would definitely say the trigger to trigger id is a convention because you assign something that differs from what it was. Anyhow, you can get rid of a conversion (no matter if you call it the second conversion or the only conversion), just use the notification id as trigger id
alias: Identify wash person
trigger:
- id: 'mobile_app_nokia_3_2'
platform: event
event_type: flic_click
event_data:
button_name: flic_80e4da786aa8
click_type: single
- id: 'mobile_app_abc'
platform: event
event_type: flic_click
event_data:
button_name: flic_abc123
click_type: single
- id: 'mobile_app_whatever'
platform: event
event_type: flic_click
event_data:
button_name: flic_xyz789
click_type: single
action:
- service: input_text.set_value
target:
entity_id: input_text.notify_wash
data:
value: "{{ trigger.id }}"
But all in all, you don’t need an extra conversion from the version you posted, you just need to move the template if-statement to the second automation and store the triggerid, which you can make the (reabable) name like you already did.