'drill bit wobble' may be a consequence of the chuck holding the bit not spinning true, although in Cinefil-san's example the possible faulty clamping of the bit may also be the cause of that effect, as suggested by one of the A2Q participants as a possibility to be checked.
Most of the uses of 芯 used in combinations (ex. 芯出し）seem to focus on the notion of centricity, technically defined in Cinefil-san's quote by the forum participant making above suggestion to the asker as "concentric run out".
In more informal style the expression proposed here seems extensively used, although it does take on an additional or different meaning in the case of 'wheel truing' -- i.e. lateral wobbling in the case of bicycle wheels, or loss of even weight distribution requiring 'balancing' in the case of 4 wheel vehicles with fast spin cycles.

The term, runout or run-out is generally used to express rotating unbalance.

In your case, the term is also used as back clearance of cutting tool (drill), therefore other terms including rotating unbalance may be used.

The text is describing the problem of the undesired shake of the drill.