Correct
Source:
https://www.chase.com/personal/credi...five-learnmore
As a practical matter, Chase will only see the credit card transaction. It probably does not matter how you make a reservation, provided that you pay at the hotel, and that the hotel's merchant account is classified as lodging.
I think the terms are a little confusing, it does specifically say "purchased through" third parties but also mentions booking direct. If you book a hotel through hotels.com and pay directly at the hotel, I cannot imagine that shows to Chase any different than booking through Marriott.com or Hilton.com