As per your link, it's only "seemingly" contradictory - that is, it emphasizes that as soon as the (previous) king dies, the (new) king is automatically immediately in office to be wished a long life, without needing to wait for any coronation ceremony. This is the rule in many monarchies past and present, though not all. So, imagine that the (old) king dies at the end of the first clause and therefore the (new) king takes office at the beginning of the second.