Is there a way to get some output back to the interface after executing a!startProcessLink(), or a way to update some variable inside the interface to insure process has been executed
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Not directly. The started process runs disconnected from the current interface, and the interface is refreshed after the user completes the other process without knowing anything.
You could persist some data to the DB and query that to know that something happened.
In general, nope - for such use cases you'd want to use a!startProcess(), or a Record Action configured to open in a Dialog.
(I've been campaigning for a long time to get "open in dialog" functionality added to the Start Process Link component, but it's in the "feature request backlog" dumpster / landfill, as far as I can tell. Which is sad because it would satisfy a lot of current corner cases.)