The Australian expressions - flat out, like a lizard drinking and flat out, like a lizard on a log, are a little easier to interpret. They clearly allude to lying flat on one's face. These are known from around the time of WWII and are recorded in Sidney John Baker's, The drum: Australian character and slang, 1959:
"Ideas of lying flatly (on one's face, not one's back) and of travelling or working at great speed are recorded in the phrases 'flat out like a lizard drinking' and 'flat out like a lizard on a log'."
They are rarely used literally though, that is, to mean lying flat out, but are just an emphatic form of the usual meaning of 'flat out'.