What's the difference between ProRes "1200", "1200 (156 lpi)" and "1200 (180 lpi)" in printing?

Clearly, the one with the highest number of Lines Per Inch (lpi) will theoretically have the best quality.

However, the practical best quality depends on a couple of other things:

• The paper used - too high a print density will end with too much ink, particularly on cheap, non-photo papers
• The source file - You will only get better prints if the source has sufficient detail. Printing a photo with not enough pixels in it may actually make the print worse at high lpi, especially if the printer tries to be "clever" and interpolate the image.

So it isn't always an obvious choice and you need to understand the source, the paper, the ink, the print driver and the printer to be able to make a truly informed choice.

Generally, you should do some experimenting to find the combination you like using and then stick with it unless using very different paper or unusual source material.

For details on this printer, check out the manuals and other information at the HP web site.

600 DPI is just that, 600 DPI.

FastRes1200 is 600 DPI with HP's special secret sauce added, i.e. variable dot size. This is in reality still 600 DPI, which however simulates 1200 DPI. According to HP, the quality is near-identical or superior to true 1200 DPI (to be honest, I can't tell a difference, either -- looks just fine to me).
Advantage: 1/4 as many pixels to render as with true 1200 DPI, thus much faster. In fact, it's exactly the same as if you used the 600 DPI setting, only just now the printer actually puts on the physical page whatever greyscales it has rendered (instead of thresholding and discarding that information!).

ProRes1200 is true 1200 DPI according to HP with constant point size, they recommend it for printing documents with minute details such as technical drawings with fine lines.
To be honest, I have my doubts insofar as 132 LPIs (or 180 LPIs) very clearly states that the resolution is not 1200x1200, but rather 1200x132 or 1200x180, respectively.

That might be because like virtually all printers, the P2055d is embarrassingly poor in terms of RAM. A single page at 1200x1200 DPI is around 18MB. So... it may very well be that a full page at 1200x1200 DPI, plus the actual document, plus some fonts is a challenge for the printer's meager 96MB of RAM. There are no RAM issues whatsoever at 600x600, so...

Personally, I'm using FastRes1200 because it looks excellent and is fast. Plus, there's no weird cheating around with high DPIs and low LPIs. Good quality, reliable, reproducable, and fast. What else do you want.