Look (once again) at this picture of the cylinder and crankcase:
[Linked Image]

The red circle shows that one port - the one closer to the crankcase cover - has thread in it, the other one does not. Briggs taps a thread in the exhaust port of most of their small engines, and screws the muffler directly into the port, because this is a cheap, reliable way to mount the muffler. In other words, the exhaust valve tappet is the one closer to the open side of the crankcase, where the timing gears are, and where the crankcase cover mounts.

Note that neither tappet is closer to the piston than the other, they are both equidistant from it, because they are lined up along the camshaft, which runs parallel to the crankshaft. The piston is almost exactly above the center of the crankcase, when measured along the crankshaft.