Barometric dampers for gas fired units do exist but are typically for oil to gas converted units or where a draft hood can’t be installed. they would serve no purpose for a two pipe 90+ installation.
I don’t know if I have given you the correct description, though.
The picture in the link you sent is pretty different from what I have.
I have a round duct that leads from the furnace (horizontally). About a foot away, there is a oval opening on the bottom of the duct. There is a free-swinging butterfly-type damper in that opening.
Is this the same thing as a barometric damper? (That is the name that one of the chimney guys gave it.)
As your unit is “converted” oil burner you probably need a barometric damper.
It needs to be properly adjusted to insure that combustion products including CO do not enter the home. This procedure requires equipment capable of measuring the level of draft before and after the damper and in accordance with the burner manufactures requirements.
My advice is to get a third opinion frpm a reputable contractor that understands oil and gas fired units.
BTW-I don’t recall seeing any that are mounted facing down as you describe.