All Spiders have Wobble. Not all Spiders show any visible indication of Wobble. Many of them go through life with no expression of it at all, no problems feeding, navigating, holding their head steady, etc. Some show no indications of it, then due to some stimulus or other develop visible expressions of it. Whether or not a snake shows a visible expression of Wobble doesn't mean that the snake is or isn't Spider.
Also, there are a number of other mutations linked to Spider that also have Wobble, but they all look different. A snake may very well show it, but not be Spider at all.