If a character has the Uncanny Dodge class ability he gains the following: "Uncanny Dodge (Ex): Starting at 4th level, a rogue can react to danger before her senses would normally allow her to do so. She cannot be caught flat-footed, nor does she lose her Dex bonus to AC if the attacker is invisible. She still loses her Dexterity bonus to AC if immobilized. A rogue with this ability can still lose her Dexterity bonus to AC if an opponent successfully uses the feint action (see Combat) against her."
Does this also apply if the character is climbing?
The rules regarding climb say: "You need both hands free to climb, but you may cling to a wall with one hand while you cast a spell or take some other action that requires only one hand. While climbing, you can't move to avoid a blow, so you lose your Dexterity bonus to AC (if any). You also can't use a shield while climbing. Anytime you take damage while climbing, make a Climb check against the DC of the slope or wall. Failure means you fall from your current height and sustain the appropriate falling damage."
While the rules for acrobatics say: "You can use Acrobatics to move on narrow surfaces and uneven ground without falling. A successful check allows you to move at half speed across such surfaces—only one check is needed per round. Use the following table to determine the base DC, which is then modified by the Acrobatics skill modifiers noted below. While you are using Acrobatics in this way, you are considered flat-footed and lose your Dexterity bonus to your AC (if any). If you take damage while using Acrobatics, you must immediately make another Acrobatics check at the same DC to avoid falling or being knocked prone."
I get the feeling "losing your dex bonus" is not always the same as being caught flat-footed, although the result is usually the same and the character has no dex bonus to AC while climbing despite Uncanny Dodge. I assume the fact that the term "flat-footed" is omitted from the Climb rules is intentional and makes a difference. Am I correct?
Being flat-footed causes you to lose your dex bonus (as well as being unable to make AoOs), but not all things that cause you to lose your dex bonus do it by causing you to be flat-footed.
Uncanny Dodge only prevents you from losing your dex bonus IF that bonus is being lost because you should be flat-footed or from the attacker being invisible.
Unlike the balancing rules that say when you're using acrobatics to move across narrow or uneven ground you become flat-footed, the climb rules cause you to lose your dex bonus without the other effects of being flat-footed (such as losing the ability to make AoOs).
I believe that you are correct in that Uncanny Dodge does not help a climber.
EDIT: missing right parenthesis
Yes. Fluff-wise, losing your Dex bonus means that you can't move out of the way to avoid an attack. Being flat-footed implies that this is because you're not prepared to do so, either by being unaware (i.e. before you've acted in combat), or by concentrating on having stable footing (e.g. using Acrobatics). While climbing, you literally can't move out of the way, since you're hanging on to something with both your hands and feet.