I am trying to understand the purpose to having engagement tokens for both Starks and Lannisters. I've looked through the rules and can't seem to find the answer.

Does it matter which side starts the engagement?

Can you only benefit from a flank if the engagement token is your faction?

http://www.fantasyflightgames.com/edge_foros_discusion.asp?efid=176&efcid=1&efidt=344144

Check 3) in the link above

Palmen said:

Checked.

3rd paragraph states about Stark cavalry in Battle#6.

The odds are that you meant to check 2nd paragraph.

Please, give us an example how engagement token type affects the battle and decision making.

So, if a unit is on a building hex, it has Cover 1, which is the same as Toughness 1. But, if there is an engagement marker showing its side, then it is considered out of the building (out in the yard attacking back) and no longer gets Cover 1.

The rulebook does a poor job of clarifying this, if it mentions it at all, but its fairly intuitive once you understand the rationale behind the rule.