the second picture your looking down onto the muffler
the exhaust goes in where it broke and it has a water
the exhaust water
steam then comes out on the rubber hose part.
IF the muffler is in good shape or bronze then I would take it off and have it rewelded.
You might consider getting a bronze or SS pipe fitting and welding attaching that fitting to the muffler. Then have a threaded pipe setup back to the motor
exhaust output off the manifold.
A lot depends on what condition the waterlift muffler is in.
SOME of these were plasticized rotospun inside to prevent rust, such as the one on my onan generator