This is wrong and directly contradicted, unambiguously by the linked paper. "For the purposes of defining the halting problem H, one should specify whether it

This is wrong and directly contradicted, unambiguously by the linked paper. "For the purposes of defining the halting problem H, one should specify whether it officially counts as halting or not if the head should happen to fall off the left edge of the tape. Although the truth of the main theorem will not depend on these details, provided we adopt a uniform answer, let us be definite and regard such computations as having not officially halted, as the halt state was not reached."