@tomson Thanks for caring! What I did: I used the visual editor, clicked on "add media", chose this image, that has a size of 920 by 165 pixels and clicked "insert into post". It is reduced to a width="300" and height="54" by the editor.
For test reasons I did the same procedure with another version of the testbanner, that is 1280 by 400 pixels. Same way of inserting it – and it shows with exactly the same width.
Now, analysing your answer, I used the text editor. Here it is shown, that the editor took another resolution and file than the selected ones. Here I could insert the direct link to the testbanner-file and show the banner in the correct size.
OK, with this workaround I know how to do it in the future. But honestly, I think that the editor should not replace the selected image with another one in a different size ...