TT rumble versus "oscillation": what's the difference?

Playing some LP's I've described forward and back movement of my mid/woofer drivers.  No obvious noise emanates from the movement that can be visualized.  I assumed that this was evidence of "rumble" but have been informed that it might instead be "oscillations".

So my question to you vinyl gurus: what is the difference between those two conditions?  Is one worse than the other?  Do both compromise performance of the speaker and the driving amplifier?

TT rumble can cause subsonic woofer oscillation which is what you’re observing.

I only get woofer pumping with a handful of warped records in my collection/system. The low cut filter in my Luxman EQ-500 works wonderfully in these rare instances to virtually eliminate cone flap. I believe it starts gradually rolling down output under 50hz and attenuates -14dB at 6hz where the effects of warps are strongest.
Oscillation is any back and forth. What you describe is usually called woofer pumping. The drivers move back and forth, sometimes a lot, but at such a low frequency you don't hear it. Doesn't hurt anything but it does waste amplifier power. 

Rumble is anything from that low frequency on up into audible bass. Rumble usually is reserved for bearing noise, although depending on frequency and level the noise may or may not be audible. 

Whatever you call it, when you have woofers moving back and forth this much it is not always easy to know why. It could be the bearing. It could also be the shelf or rack or floor. 

With an old vintage table the smart move is to first check, clean, and lube the bearing. While you are at it check and clean the belt, motor spindle, and everything the belt touches. Then once you have done this, or if you have a newer table, then move on to eliminate floor-borne vibration from reaching the table. There's a lot of ways to do this, one of the easiest/cheapest being Nobsound springs. 
Contrary to what others say, the oscillating woofers in my speakers took out one of the woofers in my Vandersteen speakers. I tried everything but nothing got rid of the problem.  I tried all kinds of footers, maple platform, bubinga platform, I even bolted a Target TT shelf to the wall, wife was quite annoyed.   This was on phono only. I bought the KAB Rumble Filter and it totally alleviated the problem with absolutely no loss of bass. The best $170 I ever spent. I have had it for at least 12 years now.