DoD by user story or by sprint result?

To meet the Sprint Goal and to respect the DOD is absolutely not the same.

If the forecast of the Dev Team is too optimistic, the Dev Team will probably not meet the Sprint Goal, for instance, the last item is not coded.

But still, at the end of the Sprint, the increment must be potentially shippable. That means what is declared as "done" is terminated.

Should the whole sprint be considered as not having met its goal, or can we just take the parts, which work, and put the button back to the backlog?

Do you think these are mutually exclusive? Isn’t it true that while the Sprint Goal was not met, some work was nevertheless performed during the Sprint? Shouldn’t the work remaining on the Product Backlog now be revised and re-estimated accordingly?

Or in other words: Does DoD apply to the whole result, or can it also apply to single User-Stories within a sprint?

The Definition of Done applies to the Increment, since that is the artifact which is potentially subject to release. There can however be multiple levels of Done which contribute to the DoD. In the case of individual User Stories this very often takes the form of acceptance criteria.

one User-Story would be to create a Startpage of a website. At the end of the sprint, everything works, except one button is not working. 

Just an observation, but perhaps your 1 user story was too ambitious?   Certainly a good subject for a retrospective discussion, but maybe there is benefit in splitting such large user stories into smaller items that are much more focused (i.e. - UX of web page, "button" stories, "link" stories, stories for other page content, etc.)?

DoD is applied to all user stories. For example, unit tested code as one of the item of DoD. Well, you could not have some user stories do not have unit test, isn't it?

Sprint Goal is applied to Sprint Result. Sprint Goal is more of validating the overview of success of the sprint (even if some of the user stories are not completed).

Does DoD apply to the whole result, or can it also apply to single User-Stories within a sprint?

Ideally DoD applies to the increment to make it potentially releasable to the customer.

From my experience what have seen is the development team defines a DONE at PBI level, which has to be adhered by all the PBIs and also another definition for increment level which is applies to the integrated components of all PBIs.

Hope it helps.