[dragonwriter]

> "I can do it cheaper" implies that I can get my costs down lower.

Zero or negative net costs are lower than positive net costs. So "I'll do it for free" (or even, "I'll pay you for the opportunity to do it") are instances of "I can do it cheaper."

> "I can do it for free" means that I'm ignoring any of my costs.

No, it doesn't; assuming rationality it means that the net costs are nonpositive, not that they are ignored (which would be irrational.)

Of course, it could be the result of ignoring some or all costs, but so could "I can do it cheaper" when there is still some charge for the work. There's nothing special about the zero charge case.