Contradiction in Terminology
It seems that "Commander" is a more general term - Commanders have an ability to lead other units in combat (not necessarily deriving from Leadership natural ability) while Warlords are Commanders who have Leadership ability and thus give units under their command a Leadership bonus. It should be noted that in some special cases Casters can give a bonus to certain units under their command (known examples are Golems led by a Dirtamancer and uncroaked led by a Croakamancer).
Duty is one of the three effects of Natural Thinkamancy (besides Obedience and Loyalty) that an Overlord exerts on all his subjects.
Duty affects Commanders only. Has Higher effect on Warlords, highest on Chief Warlord. Duty requires Commanders to use their own initiative in the service of the Ruler. Another effect of Duty is that Commanders cannot withhold information and cannot conspire against the Ruler.
Use as a Title
Word "Commander" is used as a title to address Warlords and, occasionally, Casters. In this sense words "Warlord" and "Commander" seem to be used interchangeably, although Commander seems to express more respect.
Stanley addresses Wanda as Commander, Wanda addresses Parson as Commander , Jillian addresses Webinar as Commander, Webinar addresses Jillian as Warlord, Ansom addresses Jillian almost exclusively as Commander, Jaclyn addresses Jillian as Commander , another archon addresses Jillian as Warlord.