Spindle PID Tuning question

I've read variously that basically, start with PID all set at 0 and increase P incrementally until a disturbance in the force is felt, then increase I and D until things become stable and the speed settles in a reasonable amount of time. That all sounds great. However, I've noticed a couple things in my setup that makes me hope for more detail.

Is the instability I'm looking for when I start from a dead standstill, or when the spindle has been already been running? I ask because of the following:
1) I can increase P to a point where I get a wobble (a "ringing" of sorts) in Hal Scope when the spindle turns on, but it eventually smooths out. But If I increase P more, it turns into a consistent oscillation. So, am I looking for the point where the ringing starts, or the full-on oscillation?
2) If I increase P while the spindle is running until there is instability, then stop the spindle and restart at the same speed, it will oscillate badly and not recover (at least not for as long as my patience lasts).

Similarly, when I try the Ziegler-Nichols method of increasing P until a knock on the spindle axis results in instability, there are different degrees of instability: 1) a slight bounce of the speed 2) a bit of a wobble that recovers quickly and 3) full blown oscillations. Which sort of "instability" should I be looking for?

Thanks
Darryl 

Not sure about spindles, but for servo you get to 1 and then back off till there is no wobble or oscillations.

So are you saying that when I see things like in the attached screenshot when starting the spindle, I should back off P?

The halscope is over-tuned. So if you can't pull it back under control with D then reduce the P some more.

For spindle alignment you don't really need great dynamic performance anyway, I would probably leave it quite under-tuned, aiming mainly for low final error.

Thank you both for your comments. They are very helpful.
Darryl