Yes, you write the October date because that is the date you started your UK Residency and is also the date that your 5-year count to ILR began.
When it asks how long you lived in the US, you put until October 2019 because you officially moved to the UK and activated your visa in October… which means your trip back to the US between picking up your BRP and then flying back to the UK was simply a short visit back to the US.
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