Difference between document size and print size

My best guess here would be that your printer driver is actually preventing you from properly printing your document at the right size, but we would need some additional data or test to make be sure. 

What you could do is to export your psd file to PDF (or Tiff, jpeg or soming similar for that matter). Inspect that document and see if it has the correct sizes (68 x 23 @300DPI). If so; you might wanna try and print that, to see if the issue remains.

If the issue remains, I'm positive that it is your printer (driver), if not; let us know so we can check if the issue has something to do with PPI, DPI or any other parameter or adjustment. 

"whatever i do like "use document size", or force it manually to the same parameters i have in PS, the image size is printed wrong..."

Difficult to say without inspecting options in your printer dialogue. It seems something is preventing you to print dimensions you want to print despite configuration. What it can be? Already mentioned pixel density or some option which is turned on but not easily visible to you. 

What you have to understand is that Photoshop doesn't work with sizes. It works only with pixels. Pixels have no size, they are just data points.

The print size is determined (much) later, by using the simple formula pixels per inch (ppi).

Pixels per inch means exactly what it says. There's no hidden meaning. How many pixels to an inch of paper, that's it. Pixel density determines print size. Obviously, to get good visual quality, you need a certain minimum pixel density, which is why you can only blow up an image to a certain point.

In other words, as size goes up, ppi goes down. And vice versa. But it's the same image.

Perceived pixel density depends on the viewing distance. That's why the same image can be used as a magazine spread at 300 ppi, or a roadside billboard at 10 ppi.

You seem to be mixing Pixels per inch with Pixels per centimeter 

(And it's PPI, not DPI)