DTS or Dolby Digital Pro Logic II?

I think you're confusing a few things; Dolby Digital and Dolby Pro Logic are separate entities - Dolby Digital refers to the digital audio codec that DVD, Laserdisc, and Blu-ray can use (among other things), it is usually (but not always/required to be) a 5.1 audio signal, which can be decoded and played back on its own (even on your 3.0 system, assuming things are set-up right with the receiver). Pro Logic and Pro Logic II are different things, Pro Logic is a matrix surround encoding scheme that VHS and Laserdisc (and some DVDs) can use, and it can also be applied to stereo signals (like CDs); Pro Logic II is a newer matrix decoding scheme designed to simulate 5.1 (or 6.1, 7.1, 9.1, etc) from a stereo source ("noise from every channel"). Your observations about Cinema and Music mode are correct, Music mode will put more of the signal on all three speakers (more accurately, its leaving the signal on L/R as-is, or nearly as-is, while Cinema mode is doing more re-arranging with the center and surround channel(s)).

DTS (the company) is fully independent of Dolby, and offers DTS (the codec) and DTS Neo:6, DTS (the codec) is the prime competitor to Dolby Digital while Neo:6 is the prime competitor and rough equivalent to Pro Logic II. DTS (the company) doesn't really offer anything comparable to Pro Logic.

As far as what you "should" use, thats your own choice. Neo:6's music mode should pass the L/R signal untouched to the L/R speakers, and then the synthesized (As in data pulled out of nowhere) data to the remaining channels, so that may sound more "accurate", but you're still fabricating data (more accurately: generating data from summation, it isn't magic). If I recall correctly, Pro Logic II's Music mode is similar, but I haven't seen the technical documents in a while. This isn't to say that such modes are "bad", but they are synthesizing data (they're feeding a signal into a matrix which generates multiple outputs). This can lead to some unexpected/unpredictable results with certain material, especially material not explicitly designed for such decoders. For more technical information, see the below links and their citations.

With film/television content this situation is somewhat different, as Dolby Surround encoded material is compatible with Pro Logic II, and there is actually data to "un-fold" (LtRt becomes LCRS) and then process (so LCRS can be played back via 5.1 or 6.1 or 7.1 or 9.1 depending on system configuration), this is much more ideal, and results in an actual surround sound experience.

DTS again offers nothing analogous to Pro Logic, so I wouldn't suggest Neo:6 with Dolby Surround sources, simply because you've got the PLII decoder which is designed to work with older software.

The "technically most accurate" usage will be PL/PLII on Dolby Surround sources, stereo on stereo sources and to use Neo:6 and PL/PLII as desired. There is no situation with all advantages and no disadvantages.

As far as setting up your receiver, ensure that your front L/R are set to large, subwoofer is set to off, surrounds are set to off, and the center is set to large or small depending on the speaker. With this configuration made, 5.1 sources (DTS (the codec) or Dolby Digital) will play-back downmixed to 3.0, assuming you have whatever source machine connected via digital and an actual 5.1 source; this will be much better than stereo being converted to 3.0.

Much more in-depth reading:
http://en.wikipedia.org/wiki/Dolby_Digital
http://en.wikipedia.org/wiki/Dolby_Pro_Logic
http://en.wikipedia.org/wiki/DTS_(sound_system)

Using words like "nonsense" to describe matrix processing is a bit pejorative. You may not like DSPs. But, that doesn't make them nonsense. (EDIT: I guess you agree since you changed "nonsense" to "stuff" in your post.)

Both PLII and Neo:6 intelligently steer audio from L/R to the center and surround channels. Audio that would phantom image to the center, for example, ends up anchored in the center speaker, which means the imaging doesn't collapse for people in off-axis seats. Material steered to the other channels isn't "pulled out of nowhere".

The PLII Music mode adds user controls for width and depth. I personally use PLII Music for most stereo sources, both music and movies/TV. It's all really a matter of personal preference, though.

Yes, "nonsense" may have been a bit harsh. My biggest exception with "DSPs" is that they're generally just more elegant (read: DSP-based) implementations of networks that've existed for decades. They're interpolation/scaling schemes - which explicitly means that data is being synthesized ("pulled from nowhere") based on data that does exist. This doesn't make them "bad" or "unlikeable", but it can lead to unexpected results in certain situations.

No matter how intelligent your steering is, you're still interpolating based on what you already have, and the results can sometimes be unexpected. This is not all-encompassing, and to assume that I mean all interpolation/scaling is bad would be remiss, I'm simply stating that there is no free lunch. This all applies to re-processing of mono and stereo sources; matrix sources should behave properly when fed through their respective decoder.

I entirely agree with you on it being personal preference, and have stated that repeatedly over the years - generally I don't like generalizations.

Indeed, demonstrates a gross misunderstanding of how these processes work.
Sure, for listening to 2-channel music on your 3-speaker set-up, try adjusting the PLII Music mode. The Center Width parameter will let you decide how much centre information is extracted from the recording and sent to your centre speaker.

You can adjust it for no extraction (phantom centre) or adjust it for max extraction (all centre info is cancelled from your L/R speakers and routed to your centre speaker), or any step in between. Somewhere in there you will find a setting that sounds best to you.

There is a reason why there is a PL movie mode and a PL music mode, but hey its your system.