+ 1 on the comments here. I would particularly emphasize the 1/(fl * crop factor) formula because this is what truly expresses the issue of camera shake in a way that includes the narrow field of view "reach" for crop bodies (as already mentioned, the crop factor for a FF body is 1.0). in other words, for a 300mm lens on a 1.6x crop body, the rule says that shutter speed is 1/480th sec. The only additions I would make to the good comments above, are:
1. its a rule of thumb. experiment with your own abilities -- you may find that your own personal needs are either more strict or less strict than the rule.
2. Its a rule of thumb for camer shake caused by you, not for stopping the action of moving subjects. experimentation is the only thing that will help you determine how fast is fast enough.
3. learn how to avoid shake -- lots of information out there from posture to breathing. Also learn to look for opportunities to utilize anything to help -- from trees or walls to lean against, to the the ground, or whatever, for supplimentary support when you can't use a tripod.
4. balance the rule of thumb with your expectations for the final product. Some small amount of blur might be acceptable if you won't pixel peep, you're not printing an 11x14, the wife wants it anyway because is the only shot you have with that smile, etc.
5. IS won't stop a moving subject, but it is very effective when the subject is not moving. Again, depending on your abilities and that of the IS system, advantages from 1 to 4 stops have been reported in actual practice. I've obtained some acceptable shots at 1/30th and even 1/15th second at 200mm f/2.8 (70-200 f/2.8 II IS) with my 1.6x crop body, using IS. watershed habitat birds in a zoo, for example, arn't usually in bright light, but they often sit still!
6. for moving subjects (aircraft for example, and birds in flight), you have to learn to pan. For example, following a propeller aircraft with a 300mm lens while still keeping the shutter speed low enough to blur the prop and still keep the plane sharp, takes practice. yea I prefer to blur the prop so that the plane doesn't look like its hanging from a string.
7. multiple-frame burst is your freind. In a countless number of situtions, I have combined posture, breathing, IS and multple burst, and obtained one keeper out of, well, however many it takes!
8. learn to give up depth of field for shutter speed. you did a great job of that with the snake photo. wider apertures may or may not be acceptable, but a sharp photo with a shallow DOF is more likely to be a keeper, whereas a blurry photo with a large DOF is guarenteed not to be a keeper.
9. choose ISO wisely to optimize to your goal, and learn how your camera chooses ISO if set to "auto". For example, photogaphing prop aircraft in bright sunlight requires a ISO 100 to obtain ~400th sec shutter speed and f/8 region**, and you can't always depend on the camera to make the same choice as you would. Personally I find that ISO 400 is useful in most situations with typically good results in good light. higher ISO values than 400 will give you higher shutter speeds but may introduce more noise (depending on the camera), so you have to find the right balance and know the abililties of your camera and what IQ it produces at various ISO settings in various situations. you'll need to become one with your camera in that regard :-)
BTW, the good news is that all the required experimentation is a boat load of fun.
** "sunny 16" rule. in bright sunlight and f/16, the shutter speed is 1/ISO. f/8 is two stops brighter than f/16, so shutter speed has to go up two stops, to 1/4*ISO.