That looks like a "fixed rim" sprocket to me, combines the benefits of the replaceable rim (better support of the drive links/tie straps) with the liabilities of the spur sprocket (non-replaceable). Notice it has 6 "spokes", not 7 like a normal rim type.
McCulloch and others made numerous clutch drums with the fixed rim arrangement.
The bar appears to be a "banana bar" with the safety tip that should not be flipped over, in fact there are no oiler holes on the opposite side of the bars so it really can't be flipped over to even the wear on the bar grooves.
Most of the banana bars that I am familiar with are intended for use with full size 3/8 pitch chain. The teeth on the nose sprocket are taller and come to more of a point.
Sprocket teeth for LP chain are shorter with a flatter top.
As has been pointed out, there are physical differences between 3/8LP and full size 3/8 pitch chain even though the pitch (distance between the drivers) is the same. The LP drivers are smaller and the chassis height (tie straps) is shorter, full 3/8 drivers are larger and the chassis height is taller. This means the radius of the sprockets must be different for everything to match.
There are some 3/8 LP rim sprockets available, but I've only seen them in the Stihl "Mini" spline.
The 3/8 LP or Pico sprocket is actually a bit larger in diameter due to the shorter chassis height.
Finally, the chain on the OP's saw is certainly LP (91 Oregon designation) and is the Auto Sharp or Power Sharp variety. I don't think that bar, chain, and sprocket were ever intended to work together.
Mark
