There does seem to be a bug here, Benoit. I followed the steps you outlined, with the same result.
The first update did not pre-select the attribute change in 'Inscription activité', but the second update DID pre-select it, though it did not pre-select the change to the join. After the second update, the LDM entity 'Inscription activité' contains two 'Code Club' attributes, and only the original one is referenced in joins.
Going back to your original assertion though, that [the attribute"Act_Code_club" is the same information asthe attribute "Code_club"]. Excuseme, but I must disagree - it is not the same information, "Code_club" is the identifier of the Club inherited via the rempli and possède relationships, whereas "Act_Code_club" is the identifier of the Club that is inherited via the réserve and organise relationships. I assume you expect the two relationship chains to link to the same Club, which is an integrity constraint linking the two relationships. You can avoid this situation by changing your CDM slightly - change the parent entity of 'rempli' from 'Membre club' to 'Personne'. It is a person (not a Membre) who signs up for a club activity, and part of the validation for the relationship is that the person must be allowed to do that, probably by being a current member of the club, which can easily be tested by finding the person in 'Membre Club'. That will also allow you to support possible arrangements between clubs, where a member of club A can sign up for activities in club B.