As a theoretical rules lawyer, , I note that there are several types of system:

1. Simple, crude.  These are systems where "hard skills" are prevalent and "soft skills" are often "looked down upon" or non-existent.

2. Complicated, crude.  These systems generally seem to represent a reaction to the type 1 system.  They have lots of soft skills, in some cases, soft skills for everything and aim to "level the playing field" by making sure you spend your points to do _anything_.  As a consequence they often involve lots of numbers which make it hard to keep track of everything without an external measuring device such as cards, or battleboards.  An example of this from my personal experience would be LT or CP (where you can't use a sword at all unless you've paid 1/10th of your character points).

3. Complicated, refined.  Typically these have "evolved" from type 2 systems and allow you to do anything that your "hard skills" will allow you to.  They also have "soft skills" that allow you to enhance your "hard skills" and have in game effects that would not normally be possible, e.g. magic.  I tend to prefer these systems because they present a wider scope of play for me personally then I find in either type 1 or type 2 systems.  An example of this from my personal experience would be GD or PD (everyone can do anything that they can do, although they won't do it as well as someone who's invested IC expenditure in it).

Logically there should be a 4th case which should be:

4. Simple, refined (?).  I'm currently unsure what form this system would take.  I suspect a definite move to minimise OOC concepts would be part of it.

Thus the change from so called "maths heavy" to "maths-lite" systems _isn't_ "dumbing down", i.e. catering to the lowest common denominator, it's progressively advanced system design.  I _believe_ it was Einstein who said, "Everything should be made as simple as possible, but no simpler." Something that prospective and current games designers should perhaps bear in mind.

As for cheating, well, it's possible in all systems.  I haven't devoted as much thought to types of cheating because frankly I _hate_ cheating.  The people who do it have different reasons, some can't bear to lose, others have thoughts about drama, I find it reprehensible in all cases.  The way they do it can differ as well, from the simplest just not counting hits on the hand (because that's the way they've always played it) to the rather more expensive and logistically sophisticated method of simply ordering money phys reps from game suppliers. 

edit oopsie, got carried away and forgot the important bit! 

There is - it's the one you've listed as 'simple, crude', it being a realisation of the difference between roleplay and statsplay and a reduction of the game skillset to the bare minimum.

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You gotta burn to grow!

Using the paradigms outlined above, the "complicated" rules sets seem to become refined by developing the rules and stats to enable smoother interactions, while the "simple" models instead base their refinements on the players and how they interact.  Personally, I would rather the focus of LRP development be on the latter rather than the former - another reason why I prefer simple rules sets for the kind of games I want to play.