"Honor" and "a sense of responsibility" explains very little and good, widely distributed Schelling points (koban and, to a lesser extent, commercial/transit centers information/etc desks) that both do the right thing for lost/found items and are widely believed to do the right thing explains most of it.
You are legally entitled to a reward of up to 25% of the lost property's value in Japan. Almost everyone waives it; you'll be asked about it when filling out the paperwork. In my limited experience (five times?) they do try to avoid influencing you either way with the wording, and don't say "Everyone waives this." This does come up occasionally with regards to cash hordes.
Foreigners interacting with this system are, unfortunately, frequently seen as a burden by the cops ("Can't even fill out their own forms grumblegrumble") and brushed off unless someone in the koban decides either to be a go-getter or that this particular foreigner is likely socially established. In some cases the form of the brushoff takes the form of an impromptu Japanese test ("What do you want me to do with that?") where knowing that the magic words ("I would like to file a lost property report." / "紛失物届けを提出したいです”) is treated as being suddenly worthy of prompt service.
This can be heavily YMMV based on the individual cop, the station/koban, and (regrettably) "How likely people who look like you are to be a frequent source of aggravation for cops in this neighborhood?"