Only a partial answer, but there's a Rules of the Game article that mentions this:
Normal Movement: This is not a standardized game term, but the rules (and this article) use it to indicate times when a creature uses a move action to move up to its speed -- as opposed to running, charging, or performing some other kind of special movement.
If a creature has a swim, climb, burrow, or fly speed, than those types of movement are considered normal for them. This is slightly extrapolating the not well defined meaning of "normal movement", but part three of the same article discusses this to some extent:
A creature with only a land speed can climb or swim by making an appropriate skill check, but cannot charge, run, withdraw or take a 5-foot step while doing so. That's the chief difference between climbing or swimming with a climb speed and using the Climb or Swim skill.
Since there are a set of heavy restrictions on what you can do with your movement mode, I wouldn't consider it to be "normal" movement at that point. For such a creature, it is a type of "special movement".
So a barbarian couldn't draw an axe while climbing, nor could a rogue tumble while swimming. But creatures with climb or swim speeds could! (Well, you'd still need a free limb somehow for drawing a weapon while climbing.)