| It's pretty simple, a 5.5-in rim should be used with a 180 width tire and a 6-in rim should be used with a 190.
The outside diameter of both size tires is the same so a switch won't require a rear ride height adjustment. The important difference is that the 180 is a 55 section meaning that it's height is 55% of the width cross-section. The 190 is 50% of it's width. This means that the 55-section tire has a steeper profile, it's taller.
When you mount a 190 tire onto a 5.5 inch rim it's profile becomes slightly incorrect. The too-narrow rim forces the tire's outer edges inward into a tighter curve so that you can't use this part of the tire effectively. A correct tire profile creates a correctly-shaped road contact patch essential to optimum handling, better sidewall stability with less tire flex and, and better overall tire wear.
When developing the suspension for the 916, Ducati had World Superbike racing in mind so when they sold models for the street they decided to mount 190/50 tires to 5.5 inch rims, a good combination for stable handling. It's been pointed out that WSB Ducati's then used 19/67 race tires, roughly equivalent to a 190/60 road tire. So, we got the wide tire look without the quicker turn-in handling characteristics of the 60 section race tire.
In the 1995 916 owner's manual, Ducati specified the 180/55 as an "alternative" to the 190/50 and the bike's under-seat specification sticker also listed both sizes as recommended.
It wasn't too long before buyers figured out that switching from the 190/50 to the 180/55 gave a very noticeable change in cornering feel. The 180's, mainly because of their taller, steeper profile, turn-in much quicker and easier. So eventually the word spread, and everyone who has changed to the 180's has praised its positive effects on handling.
Keep in mind that, as any street tire wears, the center section wears down more rapidly than the sides, so a 55 section tire will drop to an equivalent 50 section over the life of the tire. Consequently, the turn-in handling gets sluggish as the tire looses profile. This partially explains the rejuvenating effect that a new set of tires will have on a bike's handling, and will give you an idea as to the magnitude of the effect of switching to a taller section tire.
Most of the perceived handling difference people are attributing to size effects is due to the fact that they have installed a brand new 180/55 tire in place of a worn 190/50 on their bike.
A 180 tire is also slightly lighter. This will account for part of the subjective handling improvement experienced when moving from a 190 section tire. The weight difference between brands is greater, especially for the front tire. For example, 120/70 front Pirelli Supercorsa's (8 lbs. 6 oz.) Dunlop D207RR (10 lbs. 7 oz.) A 2 pound lighter tire will reduce rotational inertia by the same order of magnitude that you get when switching from an aluminum to a magnesium wheel.
The 190 size somewhat is stiffer because of the shorter profile. This results in increased grip and reduces the tire carcass flex (better feedback), making accelerating hard out of turns less scary. Also, if you reduce tire size, with the same horsepower you're going to stress the tire carcass more. This however hasn't been a problem, even with the most powerful street bike models.  | 
Karl
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