I think those traces are appropriate for the circuits.
In the first (grounded) the scope reference is the lower AC line, so chB will show a sine wave (the AC waveform). The bridge is grounded on the negative side, so you will get a half-wave on chC. However, you might normally expect that be a rectified wave of double the AC frequency, but recall that here the lower AC line is grounded, so only the upper line is driving the rectifier and it never goes negative. Thus the chopped half-wave.
In the second (un-grounded) the scope reference is the bridge negative terminal. Only the AC upper line can drive it, and only when it is positive with reference to both the lower line and ground (because the lower line is shorted through the bridge to ground. Again, a chopped half-wave is appropriate, I think. The scope chC will be a rectified sinewave, double the AC frequency, which is what we see.
What we can't see, possibly because of the range settings, is the effect the bridge diodes have. But that's not surprising since the AC is 240V and the diodes are 0.7V or so, and pretty much invisible unless you zoom in.