I've fixed my problem by changing the syntax to the following:

Code:
textListOfVideoIds=$(echo $uploadsOutput | sed 's/"videoId"/\n&/g' | grep "videoId" | cut -d '"' -f4)
read -a videoIds <<< $textListOfVideoIds


This is more in line with how I expect things to work with regard to redirection, and this one seems to work both at the SSH prompt and from within Synology Task Scheduler.

What I don't understand is the syntax behind the other one, and why it didn't work in the Synology Task Scheduler (but worked at the SSH prompt).
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Tony Fabris