Looks like Schaublin, from which the Habegger was derived, also have a similar fitment on the tailstock.
Back in the day such blind holes were used to hold small quantities of lubricant for centres. Often white lead. The white residue that came out when anointed with brake cleaner seems suggestive.
A loose dibber, basically a steel pin with a small round knob of suitably artistic (or not) shape on top, normally resided in the hole resady to transfer lubricant to the tailstock centre when needed.
Looking at the final picture in your post it appears there is indeed a typical form of knob showing above the top rim of the hole. The tailstocks on some of the machines on the Schaublin pages here :- **LINK** also have similar knobs showing.
Clive.