Private baths vs Hot Spring Baths

Sorry, I am confused by the different terms being used in the website.

Does private baths means "hot spring baths" that are private in the room? Or do they mean normal bath room? And we had to use the public bath to enjoy the hot spring water?

And how do I travel to Hakone if I am coming from Kyoto? Do I need to travel back to Toyko station or there is a direct train ride to Odawara station from Kyoto?

Hope someone can clear my doubts =) Thanks.

Does private baths means "hot spring baths" that are private in the room? Or do they mean normal bath room?

It depends on the context, but usually it refers to a common bath that can be used privately (either by reservation or by first-come-first-serve).

And we had to use the public bath to enjoy the hot spring water?

Usually yes, but sometimes the bath on the room also comes with hot spring water. It depends on the ryokan and room.

And how do I travel to Hakone if I am coming from Kyoto? Do I need to travel back to Toyko station or there is a direct train ride to Odawara station from Kyoto?

There is a direct hikari train from Kyoto to Odawara every 120 minutes. It is explained here:
http://www.japan-guide.com/e/e5206.html?aFROM=2158_5200