The long thing is a cylinder, the piston is the part inside that moves back and forth.
trunnion mounts are hard to find because they are not commonly used in consumer equipment. i.e. not enough volume for the NH or surplus center places to sell. thus they are typically industrial quality low volume high buck stuff. Usually the trunnions are machined, and they mount in machined pads not stamped mounts like shown.
Reason for using trunnion mount is that the effective length from trunnion to extended rod end is half the length of a closed end clevis mount, thus less tendency to buckle the rod as a column in compression. They can also make shorter tighter arcs following motions like a crankshaft, but that is not an issue on a logsplitter.
IIRC, that splitter shown uses one port of very thick wall steel as one trunnion, and welds the same thing on the other side as the other trunnion. Neat engineering. The little Brave unit uses a trunnion mount like shown, and has a second trunnion set near the closed end. the rod end set is lifted out of the bracket, the entire cylinder/pusher moved to the left, and the closed end trunnion drops into the bracket. Thus, the stowed position is 20 inches shorted than the working position so great for storage.
I am trying to build the same concept using a cyl with flat plate mount on the rod end. Surplus Center has a 3 x 19 cylinder cylinder with flat plate mount on the rod seal end, I think mine was about $120. Granted 3 B is small, but for a very light portable unit it is working out well (in progress slowly). Would that work for you? I'd normally go 4 inch Bore for any trailer mounted unit though.
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