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I believe the reason this happens is due to lazy evaluation. The data passed into geom_tile/point gets stored, but when the plot is printed, it grabs Points$x from the current environment. During the loop, this points to the current state of the Points data frame, the desired state. After the loop is finished, only the second version of Points exists, so when the referenced value from aes is evaluated, it grabs the x values from Points$x as it exists after the second evaluation of the loop. Hope this is clear, feel free to ask further if not.

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The problem arises due to your use of Points$x within aes. The "tl;dr" is that basically you should never use $ or [ or [[ within aes. See the answer here from baptiste.