[X] I killed it (Lay public claim to having killed a Walker with all the attention and the fear that engenders)
This is not a secret anymore either way, since the other option explicitly states that "those with the senses to feel it will know it".
Meaning at least some of the elders around will know it and will try to use it to their advantage, as is their nature.
So not claiming to kill it carries, in my opinion, most of the negative consequences of claiming the kill, since people we should be concerned about will know about it and use the knowledge anyway, without the advantages of claiming the kill, showing that we are not afraid of this knowledge to come out and generally being even more ridiculously dangerous to deal with than already thought of after this performance.