Help identify please: power or manual gearbox?

Yes it's a PS box you have...and the engine pictured is not a 360 by the balancer,possibly a 273/318/340...most likely a 318...

The OP will need to take a closer look. Later LA 360's had a recess in the balancer/damper ring as opposed to the cast-in counterweight that stuck out. That one may have a recess. (Enlarge the pic for a better look.) Checking the casting numbers on the side of the block is a sure-fire way to find the engine size.