If i were to see "draining board," i would assume that the writer means a place where dishes which have been washed and rinsed are put while they dry.
This is a picture of a draining board which is placed next to a sink, and upon which a dishrack would be placed to hold the washed dishes as they dry.
This is a dish rack into which dishes are put when they have been washed and rinsed, and which would be put on a draining board.
In this picture, the baby is sitting on a draining board made of steel, which is built-in, a part of the counter which holds the sink.