The Wi-Fi cell phone is the predecessor of the modern smartphone with all of its connectivity options. After the invention of the first handheld mobile phone by Motorola's Martin Cooper in 1973, cellular phones quickly became smaller and integrated more technology advances; the boom of the Internet and the creation of the Wi-Fi standard in the late 1990s eventually led to the integration of cellular phones and Wi-Fi in the mid-2000s, also partly due to the introduction of VoIP in 2004.
A Wi-Fi cell phone is a phone that makes calls via Wi-Fi, which is based on the IEEE 801.11 standard. Wi-Fi phones from the mid-2000s were tied to carrier VoIP subscriptions very much like regular cell phones, and they made calls entirely using VoIP. They could call other VoIP users and cellular subscribers alike, depending on the provisions of the carrier. Because Wi-Fi was not very prolific in public back then, Wi-Fi phones were usually used at home or in the office where wireless routers were typically installed. A phone may have both Wi-Fi and cellular connectivity, switching between the two where necessary; so if a phone is Wi-Fi only, it makes more sense to call it a Wi-Fi phone, omitting the "cell" designation since it does not use a cellular network.
With the introduction of smartphone OSs with Internet browsers, it made sense to include Wi-Fi onboard all smartphones because of the Internet connectivity it brings. Today, no phone with a modern OS like Android, iOS and Windows comes without Wi-Fi connectivity.