WW1 Feuerwehr / Bahn / Post Schirmmützen - Opinions Please

[AlecH]

I was offered these two Schirmmützen, to me they seem identical apart from one has a cockade high on the schirmmutze between two rings of red piping, the other has a cockade lower down on the outside of the sweat band. I was told WW1 fire brigade being unsure what they were, I took only one with the higher placed cockade. Thinking things over are they really Feuerwehr ? or could they be Bahn (railway) ? or maybe Post ?, one is stamped Fernspecher (telephone) on the inside. Could members tell me what they are ? and are any cockades, badges missing from the front of the Schirmützen. Any information would be welcome.
Best Wishes

[AlecH]

Here are pictures of the second Schirmmutze

[SteveR]

Nice caps. Great shape for the period.