In 1962 there was no F-550.
F 350 was as it is now, a 1 ton. Possible to have a stake truck body. Engines would probably be a 6, a 292 or a 352.
F 500 was a two ton, much larger, 20" wheels. Again, a 6 or the 292/not sure about 352.
F600, again a two ton but with a 2 speed rear axle. 292, not sure what else.
Can't tell you much more as this is memory of dear old dad explaining it when I was 10 riding with him in his '58 F600 292 4V 4 speed/2 speed real axle which was at one time a moving van tractor, then he lengthened it and put a 14 foot stake bed with dump on it, turning it into the best farm truck in the county, at least for a year or two.
No 401/477/534 in these size trucks. See elsewhere in this thread and someone has itemized the trucks for the SD motors. (F700 & Up)
If this truck has 16 or 16.5 wheels, it is probably a F350. 20", a 500 or 600.