I have template with item and trigger:
item:
agent.ping
trigger:

{template:agent.ping.last(0)}

... and host with linked template.

I change trigger name and expression:
trigger => trigger2

{template:agent.ping.last(0)}

=>

{host:agent.ping.last(0)}

Consequences:

• the trigger in the template disappears <-- correctly
• the trigger in the host links to itself <-- not correctly
• the host has a new trigger <-- correctly

See a screenshot.