In the NYC video the proud owner of the famous Hotel Chelsea acts as though the name Chelsea originated with his hotel. Not so.
The hotel is in the Chelsea neighborhood of Manhattan. It's at 222 West 23rd Street, between Seventh and Eighth Avenues, near the Empire State Building, Midtown, Greenwich Village, and other NYC attractions.
The original Chelsea is an area of south-west London, England, bounded to the south by the River Thames, where its frontage runs from Chelsea Bridge along the Chelsea Embankment and Chelsea Harbour. The word Chelsea, says Wikipedia, means "landing place [on the river] for chalk or limestone" (Old English). Anglo-Saxon Cealc-hyĆ° = "chalk wharf".