A very important exception to "dpi is just instruction to printer about how to space the pixels". Which is very true for the printer, however to a scanner, if you scan to create a picture at high dpi, the data file will be much larger than if you scan it at low dpi.
"300 dpi" to the scanner is instruction to create 300 pixels per inch scanned (affecting size of data, but not size of print). More dpi is more pixels. But if you scan 4x6 inches at 300 dpi, and print it at 300 dpi, it will print 4x6 inches size. If you print this picture at 600 dpi, it will print 2x3 inches size on paper.
"300 dpi" to a printer is instruction to space the existing pixels on paper at 300 pixels per inch (affecting size of print, but not size of data). More dpi is a smaller printed picture. Then it would be printed at the higher dpi number (spacing of pixels on paper).
However color printers generally cannot show an improved picture at much more than about 300 dpi (printing is designed for the human eye to view). They have to try to show 16.7 million possible colors with only perhaps 4 colors of ink, so they have to dither many dots to try to simulate one color, which simply won't all fit into a tiny pixel space.
So if planning to proceed, you should try some tests, and really ought to confirm that you can actually see whatever you choose to believe.