Is a jacuzzi the same as a hot tub?

In everyday use, yes. Both are heated, jetted tubs for two or more people. Jacuzzi is a brand name that became a generic word for a whirlpool tub, the way people say Hoover for a vacuum. In hotel listings, jacuzzi tends to mean the indoor jetted tub and hot tub the open-air one, but the soak is the same.

Strictly, Jacuzzi is a trademark for one company's whirlpool tubs, so not every jetted tub is technically a Jacuzzi. In normal speech, and in most hotel listings, the words are interchangeable. We use jacuzzi generically for a jetted or whirlpool tub, and hot tub for the outdoor terrace version, simply because that is how people search and talk.

The distinction that actually matters when you book is not the brand or the name. It is whether the tub is private and in your room, and whether it is heated. Get those two right and you will have the soak you pictured, whatever the listing decides to call it.