The OA used to have a Lodge Chief patch, but that was nixed. you can still find unofficial ones though. What I did during my tenure was to get a reproduction one, affix a loop to it, and wear it as a temporary patch during OA functions.
Section officers don't hold any position withing the Area or Region. I believe Regional officers do hold a position on the Regional Committee though. Often times, a Lodge officer will hold a position within the Council, as here, we sit on the Council Committee and the Executive Board. A Lodge Chief could wear a uniform that reflects that position, and distinguishes him at the Council level. Another way we identify our Lodge Chief here is through a donation. Each year, the Lodge makes a donation of $1000 to the Council Endowment in the Chief's name. This recognizes the Chief as a James E. West Fellow, something very uncommon for a scout/scouter of that age to be. So the square not for the award provides recognition.
Now, those are some different ways that a Lodge Chief can be recognized as such. As for the comment about what loops to wear and all that one poster made, my method is to have a different uniform for different responsibilities. I have a Troop uniform, a District uniform, and an OA uniform. As a youth, I had a Troop uniform and an OA uniform. When you get to the level of involvement where these questions come up, you're going to have several uniforms anyway, so I just think it is best to have each one suit a specific need.