Hi,
A couple points here...
First, when i try to explain a signal like the second one i try to give the most concise and descriptive explanation possible from the standpoint of electrical engineering, or at the very least the one that i think is the most concise and descriptive at the time of writing subject to change only by updated theory that i know of on the subject.
With that in mind, we can note that the first signal can be stated mathematically as:
V1=Vpk*sin(w*t)
where Vpk is the amplitude of the sine (and is positive) and and the second signal as:
V2=Vpk*sin(w*t)+V0 {Vpk<V0}
where V0 is the DC offset.
It should be clear that V1 is an AC voltage. For V2 with the constraint given {Vpk<V0} that means the entire waveform is above zero, but note that it can also be written as:
V2=V1+V0
and that's exactly how a signal like that in electrical engineering would be represented, and it should be clear that if it takes an AC signal added to a DC signal to make up the entire signal, then calling it simply "DC" is not a good enough description.
For a second example, note that if we call the second signal originally given "DC", then what do we call a perfectly straight horizontal line? We absolutely MUST call the straight line DC, but if we call them both DC then we have no way of knowing that the second signal originally given has an alternating 'component' to it as well as a pure DC component.
In circuit analysis this would be found from the Fourier components, where the first signal would have zero DC and the fundamental, while the second signal would have both the fundamental and a non zero DC value.
There will be contradictions to this way of thinking because the second waveform given is all above zero and therefore the current always flows in one direction, albeit it changes level over time. But when this happens there is almost always another term used to describe this action to show how it is different than pure DC. For the example of the full wave rectified waveform with no capacitance, the phrase usually used is "pulsating DC" to distinguish it from pure DC.
Also, when charging a battery with the second waveform, the battery still charges. The difference is that it does not charge at a single rate but the charge rate varies. It will be higher some times than other times. The average charge rate would be given by the mean value of the waveform which is the DC component. This makes it seem like it should be "DC", but again we could easily show a circuit that would not respond in the same way that it would respond with a pure DC signal.
So there is a difference and there should be some way to describe this difference. In electrical engineering it is described by the DC and AC components, and in more common language it is called "pulsating DC". Think of what would happen if we decided to no longer recognize ripple voltage.
A ramping signal still contains AC components even though we dont see anything going up AND down.
There will be times when this is more or less important. For example, for a battery discharging over time the DC voltage goes down slowly. It might be 12.000v at 9am, then 11.999 at 10am, then 11.998 at 11am, etc.. We usually refer to this as DC but a more exact analysis would show that it still has AC components.
So in short i think at the very least it should be called "pulsating DC" to distinguish it from pure steady DC.