Answer: when it is used as a map configuration file.
A web map is a configuration for a Basemap and one or more feature services. These can be ArcGIS Online and/or on-premise services published with ArcGIS for Server. The Web Map allows the author to alter the symbology of the services and to control the attribute information available via pop-ups.
The function-rich interface that threw my colleague actually belongs to the ArcGIS.com map viewer, a subtle distinction since you access it by clicking on the Map or New Map buttons in ArcGIS Online. This is the client to use for creating and configuring web maps. It is also entirely possible to use this for working with a web map. For a lot of users though, this is not as focused an interface as they would want. As the ArcGIS platform rapidly evolves, the options for accessing web maps are multiplying and are cross platform.
Within ArcGIS Online there is a growing library of templates to set up hosted applications. Many of these provide additional capabilities such as terrain cross-sections and social media integration. As well as this they are used by the Esri Runtime Apps: the ArcGIS app and the Collector. The new Operations Dashboard also uses Web Maps. A detailed comparison of the different clients can be found here: http://resources.arcgis.com/en/help/main/10.1/index.html#//01w100000082000000
An organisation can create a set of appropriate web maps and then make these available through the combination of clients most appropriate to their users. They can also embed web maps in custom applications and choose to embed them in their websites, either internal or external.