Is it just computers that are called hosts? Or routers are called hosts as well?
Although in some literature a host is any system that has an IP address, it more generally refers to "end systems"
A definition of host is given in RFC1122 :
1.1.1 Internet Hosts
A host computer, or simply "host," is the ultimate consumer of communication services. A host generally executes application programs on behalf of user(s), employing network and/or Internet communication services in support of this function. An Internet host corresponds to the concept of an "End-System" used in the OSI protocol suite [INTRO:13].
Following this definition, routers are not hosts (nor switches for example).
"host(s)" is not synonymous with a type of device. Rather, it identifies a system that is "hosting" data or service(s). A router could be considered a host in the proper context. A router could, although there is more clear terminology for it, be considered a host for a specific route to a said subnet that needs the route.
I suppose I was speaking more to the vernacular of IT/IS circles I work within than official definition.
Sometimes textbooks refer to routers as hosts when talking about the network layer and its primary goal of moving packets from a sending "host" to a receiving "host". but more accurately, the goal is to send packets between sending and receiving IP addresses, which can be held by edge devices (hosts) and routers. I would not consider a router a host because although it has one or more IP addresses, it cannot run application layer or transport layer protocols.