I would like to know if Wi-Fi can be generally referred to as the IEEE 802.11 standard or if the latter is a more general family.
Wi-Fi is IEEE 802.11, the same way that ethernet is IEEE 802.3, token ring is IEEE 802.5, FDDI is IEEE 802.8, etc.
These are some of the IEEE LAN protocols, and the IEEE working groups associated with the protocols. See this answer for more IEEE 802 working groups.
IEEE 802.11 is a standard that describes procedures, limits, values, algorithms to enstablish a WLAN connection.
Wi-Fi is a brand name owned by the Wi-Fi Alliance that certifies with pre-defined tests the interoperability between all device with this mark. WiFi devices are based on IEEE 802.11.
Not every device that uses IEEE 802.11 is Wi-Fi cerified.
In Short,Wi-Fi is more of a technology name and 802.11 is the IEEE standard. There are different variants of 802.11 based on your bandwidth ,Modulation schemes etc.
I would always use the specific standard in documents and datasheets instead of just mentioning Wi-Fi.This will help to give a clear cut information to the reader and engineers as well.
The below mentioned are the types of 802.11 standards
3.1 802.11-1997 (802.11 legacy) 3.2 802.11a (OFDM waveform) 3.3 802.11b 3.4 802.11g 3.5 802.11-2007 3.6 802.11n 3.7 802.11-2012 3.8 802.11ac 3.9 802.11ad 3.10 802.11af 3.11 802.11-2016 3.12 802.11ah 3.13 802.11ai 3.14 802.11aj 3.15 802.11aq 3.16 802.11ax 3.17 802.11ay
802.11 are the IEEE specifications that implement wireless local area networks. In common parlance I think 802.11 is more or less synonomous with Wi-Fi. Note that Wi-Fi is actually a trademarked term of the Wi-Fi Alliance.
Wi-fi is a brand that many manufactures and people use because it sounds nice. Most of these use variations of IEEE 802.11 but technically don't have to.
IEEE 802.11 is nothing more a set of technical standards that should be strictly adhered to. A device that is IEEE 802.11 compatible works with that set of standards. This makes it able to communicate properly with other IEEE 802.11 devices. Most devices that implement these standards can have a wi-fi sticker slapped on them, with permission from the Wifi Alliance.
Other radios such as the Ubquiti Airmax Nano Beam point to point radios are not considered to be wifi because the use different protocols and are used in different use case scenarios. People expect wifi to be accessible from their computers or portable devices.