Let me start with Noise 4. Noise 4 is "Zero-mean White Noise". White noise means the signal power is distributed across all frequencies evenly, which is what you see in the FFT result. From the histogram of the time signal we see that the signal is almost uniformly distributed, so close to a uniform white noise process. Although it appears closer to the sum of two independent Gaussian distributions, one with a positive non-zero mean and the other with a negative non-zero-mean, approximately +/- 0.25. (The result would be a zero-mean white noise random process).
Noise 1 is non-zero mean white noise, as evidenced by the spike at $F=0$ with a similar white noise spectrum everywhere else. From the histogram we see that it appears to be the sum of a zero-mean Gaussian White Noise (AWGN) process with a non-zero mean random process. The power level in the non-zero mean process is so small compared to the zero-mean AWGN process that we are unable to really discern further properties of that non-zero part that is added except for the mean which appears as a strong component in the FFT. Also from the histogram it looks like there really is very little signal component between the two, so the time domain plot appears as it does simply because you are connecting the dots as a line plot rather than plotting just the points (try using plot(time, noise, '.') to see just the waveform points. I suspect that it is a Gaussian distribution with a mean of 10 that is sparse compared to the non-mean noise waveform.
Noise 2 has been low pass filtered, and looks like it may have possibly been a white noise process that has been filtered by adding to itself after having been delayed by one sample. This could be called Pink Noise since the low frequency noise dominates (drawing an analogy from the visible light spectrum). From the histogram we see that this is well approximated as a Gaussian random process. So in this case it would be band-limited Gaussian noise.
Finally Noise 3 has a negative DC offset. Because of the scale, we cannot see if the noise process is pink or white (would need to zoom in on the FFT to observe that.) However from the histogram we see that it is a Gaussian distribution with non-zero mean.