Hi maki,
Your understanding of OpenGL API implementation is correct.
Implementation of API differs among graphics vendors but header is same across all the vendors. That means, the header can be found on OpenGL website, but the API implementation should be downloaded from vendor site, which comes with graphics driver.
For example, if the machine has only AMD graphics, then you should download graphics driver from AMD site. If there is graphics from another vendor (Intel integrated or an NV board), then you should download their graphics driver too. The application links against "opengl32.dll" which is part of OS. That dll determines from DC of target window which driver should be responsible for rendering to it and loads that. This is the ICD mechanism.
The latest AMD driver supports OpenGL 4.4.
Thanks!