According to the Help, when using the Web Adaptor the urls for REST endpoints should be http://<web server name>/arcgis/rest/services or http://<web server name>/<web adaptor name>/rest/services.
Only http://<web server name>/<web adaptor name>/rest/services works for us; not http://<web server name>/arcgis/rest/services
Can you post the link to the help article that you mentioned?
My understanding is that arcgis is the default instance name. If you install the web adaptor and change that then you get the web adaptor name >instead<. I don't believe that you get both.
If you name your web adaptor arcgis, then it doesn't matter what you do; it'll resolve to the same location. If you don't though, then including the port or not including the port should be the main difference:
For example, if my default instance name is arcgis and I configure the web adaptor with arcgiswebadaptor:
http://<servername>:6080/arcgis/rest will work. http://<servername>:6080/arcgiswebadaptor/rest will not work. http://<servername>/arcgis/rest will not work. http://<servername>/arcgiswebadaptor/rest will work.
I guess I got confused from reading in the help "With the Web Adaptor: http://<web server name>/arcgis/rest/services or http://<web server name>/<web adaptor name>/rest/services"
