terminal: Fix Alacritty TERM handling

[liam-fitzgerald]

setup_env modifies the current process's environment, which we are ignoring in favour of the env variable. Instead, we copy over the relevant variables from the current environment after modification so that we don't have to re-implement the logic.

Closes #17991

Release Notes:

• N/A

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[liam-fitzgerald]

@cla-bot check

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[SomeoneToIgnore]

A bit surprising to see this small thing getting stale, will close this to keep the queue cleaner.