Ouch!!
You are correct, the arrow SHOULD face the exhaust port.
Also, the skirt isn't why the piston has to be installed correctly but because
the ring ends are not supposed to rotate as there are "pins" in the piston to keep the
rings from moving. Should the piston be installed backwards, the ring END GAP can catch
a port and tear it up on it's own...
Most scoring to the piston and cylinder is almost always on the exhaust side.
(Overheating, not enough oil in the mix, etc).
If it came off that way who ever assembled it last (factory or previous tech)
messed up bad.
And 80 PSI is not enough to run the engine.. The new piston and cylinder should solve your issue...
.