Umm... that's because that Phone Icon is a link directly to the Phone app that was already installed. If you were to go into contacts and try to call somebody it would ask which to use since no default is set.
nope, doesn't do that, it still uses the default app to make a call, the dialer one app doesn't even show up
You'll have to set Dialer One as the new default calling app. Poke around the settings of the Dialer One app, while in it, and see if there is a "Set as Default" option or something similar. If not, you might have to go into your Apps, locate "Phone" and clear the data (I believe). This will make the default phone forget that it has priority, and the next time you attempt to open a phone link (just google a pizza place near you and open its phone number on google maps), it should ask you if you want to open it with Phone or Dialer One (and Remember This Selection, setting Dialer One as the new default).
Also, the icon on your homescreen will still be linked to the default Phone app, no matter what. You should delete the shortcut and create a new one in its place, for Dialer One. You can usually do this by dragging the old Phone out to trash (or long pressing it and hitting delete), and then going into your Apps Drawer and dragging the Dialer One app into the new empty spot.
(OR some phones actually let you choose what app that icon will open by longpressing, choosing "edit" and browsing your list of apps for Dialer One... if you really like how the default Phone icon looks or something and want to keep the style but have it open Dialer One)