A question that game up in our gaming session today (Of Leaves and Stewed Hobbits - we got as far as the caves):

There are two major ways to use traits: a) to pass a test automatically that is related to the trait, to gain advancement points if you suceeded in a roll and invoke a related trait.

Now to my mind both of them apply to the same situation. So is it then the player's decision if they want to pass the test without risking to roll, or take the risk, roll, and gain an advancement point (loremaster agreeing?).

The example from the book details a roll on athletics, after which the trait 'Bold' is invoked. Now if the trait is related at all, it could have been used in the first place for an automatic success.

How do you handle this in your sessions?

You have it right. The player can either go with an auto success waiving any possibility of a great/extaordinary success OR roll and with success explain how the trait played a role in the success (this is the invoking of the trait that it references in the rule book). If the player has not already gained an advancement point in that skill group the invoking of the trait gains him his first. Again invoking specifically means narrating an explanation how the trait played a role in the character's success