I'm thinking along the same line as Gary123: The symptoms indicate a poor connection between a battery terminal and the clamp on that terminal (maybe both terminals/clamps). When you try the starter, the current causes a big voltage drop across the poor connection -- the voltage on the wires would drop way down, even if the battery were OK.
An easy way to test it (to make sure you have to pull the clamps and clean everything up): while trying the starter, measure the voltage at the battery terminal posts. That is, on the posts themselves, not on the clamps or any other part of the wiring. (Note that with "modern" batteries that have screw-in connections rather than clamps, this test cannot be done.) If the battery voltage stays up where it should be (about 12-point-something volts), then you have a poor connection. You could even measure the voltage between each battery post and its clamp to try to see which connection needs fixed.
And, yes, the corrosion on the starter cable MIGHT BE related to the voltage drop problem. Corrosion there could imply corrosion on the terminal/clamp mating surfaces. But corrosion on the cable indicates another risk: it could mean that the wire is corroding through. If the wire does corrode through -- or even only most of the way through, for the starter cable -- you will lose battery power. Check the wire, and (if needed) fix it before it breaks.