That Acer H213H has DVI and HDMI - they're both digital so they're absolutely identical in terms of quality.
DVI however has less issues like the one you mention, so best thing for you would be to get a DVI cable which is about 5-10$ and worth it.
Now you may be able to adjust that problem you have by going in the Control Center and it's somewhere under
Go in the ATI's control center and see what's there under Digital Panel 1 > HDTV Support or around that section (I have both monitors on DVI so I can't tell you exactly) for something which let's you "overscan" or "underscan" the signal:
Maybe it's something like this:
By default, the driver may output something like 1850 x 1000 (something like -7% in the picture) because it assumes some TVs may have borders that would block the pixels on the sides so it would annoy you.
So set the resolution back to 1920x1080 60hz and move the slider.
When you set the ouput to 1680x1050 you basically disable this feature so that's why it works, but the image should be of worse quality, blurrier.