Why do some melons contain removeable dark tiny bits? Here's my procedure for cutting melons:

Wash the original whole under tap water.

Peel/unpeel the rind.

Wash the peeled melon again under tap water.

Wash my cutting board and entire knife.

Cut the peeled melon.
For convenience, call these tiny bits 'motes'. Possibilities for their origin:

Growth inside. But how can they originate from inside?

Originally on rind? They might've fastened to the knife while I cut the rind, and became transferred to the inside. But this looks unlikely: see 4 overhead.
Not growth inside
As you point out it is highly unlikely that these flecks are growth already in the melon when you cut it for a couple of reasons: 1) there's no wound shown in the fruit into which the contamination might have been introduced and 2) it seems you are finding them only on the cut surface, and the odds of cutting the fruit exactly where these very few and small bits happened to be are poor.
Unfortunately, after ruling that possibility out, it's difficult to know where else in your environment these bits might be coming from. The knife is the best possibility, as you guessed, so is your work surface, or your hands, or these could be tiny bits of ash from the air if you have a wood-burning stove, and so on.