Create a new context for template triggers

[doug-hoffman]

Proposed change

Create a new context for template triggers referencing the old context, if available, then fire template_triggered event. This is the same approach taken by automations and ensures that we always have a valid context to pass along when triggers don't supply one.

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Deprecation (breaking change to happen in the future)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes Trigger Update Coordinator: Running script requires passing in a context #99914
• This PR is related to issue:
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
• I have followed the development checklist
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• The code has been formatted using Black (black --fast homeassistant tests)
• Tests have been added to verify that the new code works.

If user exposed functionality or configuration variables are added/changed:

• Documentation added/updated for www.home-assistant.io

If the code communicates with devices, web services, or third-party tools:

• The manifest file has all fields filled out correctly.
  Updated and included derived files by running: python3 -m script.hassfest.
• New or updated dependencies have been added to requirements_all.txt.
  Updated by running python3 -m script.gen_requirements_all.
• For the updated dependencies - a link to the changelog, or at minimum a diff between library versions is added to the PR description.
• Untested files have been added to .coveragerc.

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[home-assistant]

Hey there @PhracturedBlue, @tetienne, @home-assistant/core, mind taking a look at this pull request as it has been labeled with an integration (template) you are listed as a code owner for? Thanks!

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[github-actions]

There hasn't been any activity on this pull request recently. This pull request has been automatically marked as stale because of that and will be closed if no further activity occurs within 7 days.
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