How to Specify and Tolerance a "Sharp" Point

I need to have an electrical probe machined; but I am unsure how to specify my requirements. I am an electrical engineer with only moderate mechanical knowledge.

The probe will be machined from 0.125" diameter stainless steel round stock. The last 0.25 inch before the point necks down to a smaller diameter - maybe 0.062" ?. The exact shape is not that important.

My question is in how to specify the pointed end of the probe. I want it to be sharp enough to dig in a little to the circuit under test but not so sharp that the point is fragile. Also, I don't want to specify overly-tight tolerances and push up the cost.
I am assuming a Swiss screw type machine will be used - but I know nothing.

What is "sharpness"? Is it the angle leading to the point or is it the diameter at the very end?

My initial stab at it would be something like: "a cone with 60 degree angle sides". But this seems to imply an infinitely small point.

Should I specify a flat end of a certain diameter? (i.e. a conical frustum instead of a cone?); would 45 degree sides be better - more durable?

As for sharpness, an ordinary pin (hatpin, safety pin, needle) works for me. I keep a few in my meter case, to take readings from tiny terminals, and to puncture insulation, etc. I grasp them in crocodile clips.

You want the point to be sharp enough to dig into metal a tiny bit. Otherwise if it's dull, the point slides along metal surfaces.