Hello Nicolas,
Is it possible to trigger an external API request to update another directory whenever an AD user changes his password?
Yes, it is possible using a PowerShell script in a Business Rule triggering After resetting password of a user. In the script, use the %unicodePwd% value reference to get the new user password.
Even if the password change didn't occur using adaxes, but directly in Active Directory?
Unfortunately, there is no such possibility.