Two
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No. Three points do. Two points determine a line.
Two. Two points determine a line. Three points determine a plane.
It takes exactly 2 distinct points to uniquely define a line, i.e. for any two distinct points, there is a unique line containing them.
2 points
Any three points will determine a plane, provided they are not collinear. If you pick any two points, you can draw a line to connect them. An infinite number of planes can be drawn that include the line. But if you pick a third point that does not lie on the line. There will be exactly one plane that will contain the line and that point you added last. Only oneplane can contain the line, which was determined by the first two points, and the last point.