Not a big fan of using head motions or type of spindle to define the difference, as there are lots of exceptions to each. Many 5-axis mills move the head in various ways beyond just Z. Datron makes CNC mills (excellent ones, BTW) that look like these CNC "Routers" but they call them "Mills".
Some refer to "Gantry Style" mills as routers. The "Gantry" is that travelling bridge with the head on it. But, Some gantries still move the table, often in just one axis, usually Y.
There are CNC "Routers" that have very expensive high speed spindles with toolchange capability. They're not just lashing a woodworking handheld router onto the CNC.
In the end, "Router" seems most often to refer to a gantry-style machine that is most often used to cut wood and soft materials like plastic or foam board. "Mill" seems to most often refer to machines that are usually not (but can be) gantry-style that are most often used to cut metal and harder materials.
Personally, I prefer to use the terms interchangeably and realize that you can use one to do the other's job in many cases.