Wfi is, by itself, only a connection between two devices which can then communicate with one another.
The point is, that the term "Free WiFi" usually refers to a setup where one of these two devices is, among other things, more or less directly connected to what is nowadays commonly referred to as "the internet": a system which does allow more or less unfiltered communication with almost any endpoint reachable by said "internet", where the word "Free" also indicates that this setup is actually (cost-)free to be used by any WiFi-capable endpoint within reception range of the antennae involved.

Hence: No. "WiFi" is neither "internet", nor "a connection to a private internet connection". Wifi is, at best, a connection to a private network, which happens to be quite often but not necessarily everytime configured such way that the connecting party is allowed to use the private network's connection to the internet.

However, since "the internet" does in itself not exist at all as it is in the end just the sum of all the interconnected private networks from the big intercontinental carriers down to the private networks of apartment renters, hotels and coffee shops, every such private network being connected to that system of interconnected private networks is by definition actually a part of "the internet", it is therefore indeed correct to assume that an offered free-to-use WiFi system is in fact a free connection to the internet and thus "the internet".