I struggle to understand how someone could cut themselves while applying tape.

Cut tape to length,
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apply to razor with blade facing up away from hand. If it is short cut another - you will know before applying.
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Smooth down from the spine (note the tape is not smeared down tight in the middle).
When starting to round down over spine start in the middle and work outward till it is smooth and tight.

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It is also much easier to center this way.