The 6150 won't come anywhere near the performance of the 1600. Depending on the game, it could be imperceptibly worse or better than your 9200. Even a 10% performance difference translates to 1-2 frames/sec at that level.
The 9600 is about the lowest, utter bare minimum for running modern games. The volari may also be acceptable, but who knows re: quality of drivers and their compatibility with all games.
Let's look at 3dmark05 numbers. That article quotes 700 3dmark05s for the 6150. This translates into 18.6 frames/sec in far cry at 800x600, and 12.6 frames/sec in doom3 at 800x600.
Unfortunately the 9200 won't even run 3dmark05, and is too old of an extreme budget card to be reviewed with new games like far cry and doom3. Most enthusiast sites only review the extreme budget cards when they are new, they correctly figure 15 frames/sec at 800x600 in modern games results are interesting to no one.
Anyway, the 9600Pro pulls about 1800 3dmark05s, so you can see how it's got significantly more oomph than the onboard card. The 3dmark scores won't perfectly translate to to a doubling of frame rates in games, but I've come across people claiming 25-32 frames/sec in far cry, and hardocp had the card pulling 32 frames/sec at 1024x768 in doom3. So 3dmark05, for all its faults, is an okay indicator of relative performance.
To give you an idea, 6600GT and X800 class cards pull between 3500 and 6000 3dmark05s. These are now considered minimal for acceptable gameplay with today's games. Less than 3000 3dmark05s is only suitable for playing bargain bin games from 2000-2003.