I'm not finding enough references to tell either way. Is one a "direct pull"? And the other a single tube with a sideways/sliding pull? What's that style called?

The 803 and 804 used the exact same footboard casting. The 803 came first, around '57 and the 804 was a new and improved model, showing up in the 63A catalog. It is not a stamp, but is actually part of the mold for the pedal. There was no reason to have separate molds for two different models that used the exact same part. That number on the pedal casting is there more for the workers in the factory, than for the customer.

You will never find a 804 with an 804 casting number on the pedal, because they never made them. Any of the 804's you have seen will be correct.

Edit: I went back and checked for later Slingerland hi hat models on ebay and every 806 Dynamo auction that showed the bottom of the footboard showed a part number of 803 as well.

Congratulations on picking up those drums!

As an aside, many of us have tried the flat-based hi hats in order to assemble a legit vintage set, only to discover that the flat-based stands can really wobble. Depending on your style, and I usually play with my heal up and the ball of my foot bouncing on the pedal between pedal presses, the stand can wobble A LOT.

I certainly don’t want to discourage your search. Just giving you an old-fashioned heads-up.

Josh