It seems quite unlikely that templar abilities are a type of magic i would agree, it would seem from talking to Alistar that the lyrim is not needed and may not even help. Though the fact that lyrim supposedly increases their talents power could lead you to believe otherwise. Though arent spirit warriors using magic since they have a special connection to the fade in a similar way mages do?
Gaider himself said what the templars do is a form of magic. And if you compare their abilities with the school of spirit, they are almost identical. Only templar powers are slightly more diluted and less powerful than a mage practicing the school of spirit. If I were playing a mage warden, who became an arcane warrior and mastered spirit magic, he would be an overpowered templar.
What the templars do are a form of magic, and there are spells that mages cast which can only affect another mage. So the templars are using magic. Just slightly different from the mages and their connection to the Fade.