I cannot get it to show on another MacBook.
I'm expecting the list to be shown when I press Control+Space.
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The issue is that often conflicts exist between the shortcuts used for changing Input Sources and those used for showing the Spotlight search field.
To check/change these, follow these steps:
Note: Obviously you can change either or both sets of shortcuts to suit your needs, and you may need to do this on either or both Macs.