Not familiar with the Classic, but the 737NG Boeing Maintenance Manual (22-11-01) has this to say:
"Single Channel Flare
Automatic flare is for dual channel approach, however, since it is part of the A/P design, it can function during single channel approaches. The flare mode does not annunciate on the FMC in a single channel approach. The autopilot should be disengaged manually when the airplane reaches the decision height"
Re 4Screwaircrew's quote:
"Full automatic approach and touchdown capability are not available. FLARE is not annunciated, and stabilizer trim bias is not applied.
An A/P go-around is not available."
The two quotes can probably sit happily, side by side, without too much fiddling
"Full automatic approach" probably means that the autoland has all the aircraft's safety features operating, such as stab trim bias, go-around, etc.
The Classic may or may not be similar to this.
Rgds.
NSEU