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The question does not ask for an algorithm.
The question is in the following excerpt: "I suspect, or rather I'm hoping, that both curves, the one with the smallest segment count and the straightest one are identical. I'd appreciate help showing that is or isn't the case." Also, Bill Thurston asked in a comment, "Is it correct that you are asking whether there is a path that simultaneously minimizes the number of segments and also minimizes the sum of the absolute value of angles?", to which the reply was "yes".
You have indicated in a comment a solution in the case that the polygon is not simple. The question remains unanswered in the case that the polygon is simple. The original question made no indication of this assumption, but it appears to have been the intent. So the question may be treated as follows:
"Suppose a simple polygon is given, along with two points, s and t, within the polygon. Is there a polygonal path from s to t that simultaneously minimizes the number of segments and amount of turning. Simple means homeomorphic to a disk; a polygonal path is a piecewise-linear curve; and turning is measured by the sum of the absolute values of the turning angles."
I believe the question as I have written above is appropriate for MathOverflow. I would have gladly edited this into the question long ago, but I don't have enough reputation to do so.
(Err, the path must of course remain within the polygon, which is closed -- that is, includes its boundary.)
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