There's another way of checking the alternator type, in the Briggs Repair Manual for the OHV singles.
Start with this ID chart, and see what you can see.

[Linked Image]

Of the most likely options, the referenced Figures 10 [single pole connector] and 11 [two wire, two pin flat connector] have a single diode in the connector; Fig 12 [single wire into a two pin round connector] has two diodes in the wiring harness. Fig 13 has an external regulator/rectifier unit, so doesn't look likely.

For sure, finding Briggs info without at least the right Model and Type numbers is painful. Pity about the rust.
In that model number, xx1587, the 8 doesn't fit. As grumpy says, the first two digits will be 28 or 31, for that engine model. The other numbers, taken in order: 1 is feasible; it's the basic design series. So is 5, as 5-9 here are Briggs' vertical crankshaft designations.

Then the next one - 8 here designates 'Plain output Bearing, Auxiliary Drive (PTO) Perpendicular to Crankshaft'. Which is not what you have! The most likely numbers here are 0 [plain output bearing, splash oiling] or 7 [plain bearing, pressure lubrication].

7 makes sense as the final digit; it means 'Electric Starter, 12 or 24 Volt, Gear Drive, with Alternator'.


And as grumpy suggests, a thorough cleanout of the fuel tank and carby is indicated, if it hasn't been done.
Water contamination is highly likely, given its history, and will cause trouble down the track, if it isn't already doing so.

Cheers,
Gadge

"ODK Mods can explain it to you, but they can't understand it for you..."

"Crazy can be medicated, ignorance can be educated - but there is no cure for stupid..."