Hello Patrick
A WAN is a network that interconnects remote networks. In this context, the Internet itself can be considered a WAN. It however is not a private WAN.
A private WAN is a service that you purchase from an ISP that delivers interconnectivity between your remote sites over an infrastructure that is separate and independent from the public Internet. Some examples of private WAN services include MPLS, Metro Ethernet, Frame Relay, or leased lines.
On the contrary, VPNs are not private WANs because you typically run those over the public Internet. SD-WAN is not a private WAN either, but it can leverage both public and private WANs as part of the connections it manages.
I hope this has been helpful!
Laz