Hi Barbara,
The timetable on our website is for the small LOCAL ferry companies connecting KOS and KALYMNOS ONLY. Their schedules are not on the websites you visited, because these only feature the large ferry companies connecting many islands to larger ports, like Piraeus or Rhodes. They cannot possibly feature all the local connections between smaller islands. So, the ferry timetables on the websites you saw are in fact correct, but they are not the only ones available. They are just on larger ferries.
Also, keep in mind that the larger ferries, like Dodekanisos Seaways and Blue Star, depart from the main port of Kos Town, whereas the local ferries depart from Mastichari, a smaller port. Mastichari is much closer to the Kos airport and much closer to Kalymnos.
The timetables on our site are also correct as far as we know; we get them directly from the local ferry (although sometimes they make changes that we’re not always notified about).