Hi Need. Companies like AOL, Juno, NetZero and Earthlink offer Dial up services they call High Speed Internet. For example, this is from NetZero's home page:
NetZero HiSpeed Internet offers accelerated dial up Internet access, with Web surfing up to five times faster than conventional dial up and includes Pop-up Blocker. The NetZero High Speed Internet service works from any phone jack using a standard dial up connection does not require any new hardware.
In the states, Cable and DSL access (along with T-1 lines) are referred to as broadband.
The term
wireless internet is ambiguous. It could refer to 802.11 communications, but these are generally referred to as Wireless networking or WiFi.