Not bad at all. You said pretty good, and happy, and I'd agree, so not sure of any question? I suppose asking why it wasn't bad as before? I don't know enough about the two situations to venture a guess. Assuming same overhead fluoresecnt with bounce flash. It could be possible that you randomly caught the flicker at its best point (although I don't detect much color difference issue from the flash). Or maybe the flash exposure was strong enough to overwhelm the ambient exposure so that fluorescent didn't matter? Or maybe vice versa?
If you want to discuss it, how was white balance done?
I can see that much of the light is from overhead, which could be bounce, or or it could be ceiling lights, or both. I can't tell the difference. Because the color is better, I'm guessing it is mostly flash, and ought to be good. But I can't tell if flash is main light or just a low fill level.
You are getting a pretty good frontal fill, which seems stronger than the catchlights in the eyes would suggest it should be. Was a bounce card or dome used on the bounce flash? I'm thinking not, and that makes me think the overhead lights are dominant and flash is just fill? But I don't know, I wasn't there.
If I were wondering and had opportunity, I'd try some more test pictures. Don't need the family for a subject, or even the background for this, just need to be in the same room with the same lights... a photo of a chair or something would work, if all else was the same. Put a piece of plain white copy paper in it to judge color. Use exactly the same location positions and camera settings, but a picture WITHOUT the flash would tell us how much the fluorescent overhead lights were contributing. It will either be a dark picture, or reasonably exposed (at same manual settings), depending on how much contribution from the overhead lights. If pretty dark, that's why the flash picture was good, fluoresents were not much factor.
Or if a significant old style fluorescent contribution, we already know 1/160 second is no good, so then also try 1/60 (better) or 1/125 (probably pretty good) (if assuming North America). The camera actually implements 1/64 and 1/128 for these shutter speeds, but the power is 60.0 Hz. So close, but not perfect, and anything else is Not good.
Because 60 Hz electrical current passes through zero voltage 120 times a second, and these lights react. 1/160 second is 3/4 of one of the 120 Hz cycles, and this 3/4 cycle is randomly positioned every shot. The fluorescent color will vary 120 times a second with magnetic ballast, randomly with respect to your shutter. Slowing the shutter to exact multiples is better, because the partial cycle captured will be minor compared to few more complete cycles also captured.
The best way to fix magnetic ballast fluorescent lighting is to NOT use it for photography. You can replace with electronic ballasts to eliminate the flicker, but you would still have fluorescent lights.
Hold this shutter speed but do any proper exposure, without flash. Don't use Auto WB, which will just confuse and hide things. The color should be better, and repeatable then in every subsequent test. Take 5 or 6 of same, to assure there is no variance now. Later when adding flash, overhead color will still different than flash color though. I suspect you really want to use one or the other lights, but not not both.
When adding flash, 1/60 second is fine if using flash, flash is faster and does not care what the shutter speed is. 1/120 will keep out more of the fluoresecent, which is then likely a better color without it, since it won't be the same color as the flash.
I would also try another test (of the chair, etc) with overheads turned off, using only the flash (same settings). I seriously doubt that one will be a black picture. With these tests and comparisons, you will know more about the situation, about what ratio of each you are dealing with. Or a light meter would do much of the same, except for color.
