"How important is it to use the actual hotel website instead of the variety of other website booking possibilities"
You'll get a variety of options on this. I'm on Team - Book with the official hotel website. Others like Booking.com because they can keep all their reservations in one location.
I book with the hotel because I prefer not to have a middle person between them and my reservation. SOMEtimes if you book direct you can get free breakfast or a small percentage off. ALSO look to see if the hotel has a FB page where you might see a deal on the price.
Whatever you do, DON'T be in a hurry when you are making reservations. It's easy to google a hotel name but sometimes you are put onto a travel agent site where you will get extra charges. IF you book someplace like Elizabeth's Holiday Inn location then you can go thru the official Holiday Inn website which is actually IHG. IF you google, be sure to put in Official "X" Hotel website.
I recently was careless when looking something up on a hotel website I've accessed before on my computer and just went to the first website that came up. As soon as I got there I knew it was wrong because it did not have the "right" picture that the hotel uses for its website. If you were unfamiliar with the hotel's website you'd never know although the url was not correct when I looked at it.