You are correct that the full serial number is found on the front of the frame and underneath the barrel. The last two digits are found on the bottom edge of the trigger plate, the bottom edge of the takedown lever, and the rear face of the toggle. All the other numbers require disassembly to be seen.
The 4 3/4 in. barrels are commonly cal. .30 Luger. The four-inch barrels are expected to be 9mm. 9mm 1906 AE are somewhat less common than .30 caliber.
--Dwight
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