I think YouTube or Vimeo does not have (even paid) any longer API, that will allow me streaming with use of the progressive download the video files (MP4, MPEG-TS or even FLV) I have uploaded there, right?
Is there possibly other platform have such API?
Petr
First of all, according to API 3 Link to video not showing Vimeo is now handling API support here with additional support via Email.
For a PRO account, here are the steps to accessing MP4 files for your videos on Vimeo using the new V3 API:
This will return a "files" key that includes multiple video sizes.
If you're not a PRO user or are attempting to access files uploaded by other users, this link seems to do the trick: http://player.vimeo.com/v2/video/VIDEO_ID/config. This only works if the video is downloadable.