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To interpret a regression interaction simple slopes plot comparing low, mean, and high levels off the moderator:

Does where the line intersect matter? If lines intersect closer to 0 on the x axis do you interpret this as an interaction pattern among participants with lower scores on the predictor variable (x)?

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The point where two lines intersect gives the value of the predictor where the fitted response to the predictor among the two groups (whose two lines are intersecting) is equal. Whether this is a point of interest depends on your application.

If lines intersect closer to 0 on the x axis do you interpret this as an interaction pattern among participants with lower scores on the predictor variable (x)?

No. An interaction is a global phenomenon (unless you are using a spline or similar transformation of your predictor). It just means that the two groups' fitted response to the predictor $x$ is equal for low values of $x$, and therefore not for high values (again, assuming you are not using a transformation).

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