You can't argue with a black object and a white object alone, as I think you partially understand in trying to build your thought experiment. You need a little bit more to define things properly. See whether the following helps.
Imagine a black object at a temperature $T_0$ and a white object also at $T_0$ inside a perfectly isolating box full of blackbody radiation at some higher temperature $T_1>T_0$ (i.e. without the black and white objects, this radiation is in thermodynamic equilibrium).
To understand exactly what would happen, you would have to describe the "colour" of your objects with emissivity curves that show emissivity as a detailed function of frequency. So your "black" and "white" would need to be defined in much more detail. You would also have to define the surface areas of the two objects and what they are made of (i.e. define their heat capacities). But all of this only effects the dynamics of how the system reaches its final state, i.e. these details only influence how the system evolves. What it evolves to is the same no matter what the details: the box would end up with everything at the same temperature such that the total system energy is, naturally, what it was at the beginning of the thought experiment. "Blacker" as opposed to "Whiter in this context roughly means "able to interact, per unit surface area, with radiation more swiftly": the blacker object's temperature will converge to that of the radiation more swiftly than does that of the whiter object, but asymptotically the white object "catches up". Blacker objects absorb more of their incident radiation its true, but they also emit more powerfully than a whiter object at the same temperature. The one concept emissivity describes the transfer in both directions. Think of emissivity as being a fractional factor applied to the Stefan-Boltzmann constant for the surface as well as being the fraction of incident light absorbed by the surface relative to a perfect blackbody radiator.
This description is altogether analogous to that of the situation where $T_0<T_1$. Begin with $T_0=T_1$, and you've got thermodynamic equilibrium from the beginning. Nothing happens, of course.
Maybe the following will help thinking about what is a really quite a complex question: it would be a fantastic last question for an undergrad thermodynamics exam BTW: You can abstract detail away by saying lets define object $A$ to be blacker than object $B$ if, when both objects are made of the same material, are the same size and shape, the temperature of $A$ converges to the final thermodynamic equilibrium temperature more swiftly than that of $B$ when they are both compared in the box-radiation-object thought experiment above.
Thinking about this now, I am not sure whether the above definition would hold for every beginning temperature of the radiation. Maybe there are pairs of surfaces whose relative blackness is different at different beginning temperatures such that $A$ is blacker than $B$ with some beginning temperature whilst the order swaps at a different beginning temperature. I think it is unlikely, but that is probably a different question altogether.
By the way, which pub do you drink in? I might come along.
Afterword on a Heater's Colour:
You ask by implication what is the best colour to paint a heater. This is not a simple question and involves the dynamics of the heater system. It's really an engineering question. I suspect in general it is better for them to be blacker rather than whiter. Here's a glimpse of the kind of factors bearing on the situation.
If you can say a heater has a constant nett input of $P$ watts, then at steady state that's going to be its output to the room, altogether regardless of its colour. There may be a materials engineering implication here: if you paint the heater whiter, and if its dominant heat transfer to the room is by radiation (rather than by convection or conduction), then it has to raise itself to a higher temperature than it would were it blacker so as to radiate $P$ watts into the room. So its materials might not be as longlasting, and it might be more of a fire hazard than it would be were it blacker.
If the heater is the hot water kind, and again if radiative transfer is significant, then the heating system has to run hotter to output power at a given level if the heater is whiter. At a given flow rate and given temperature of heating water, the heat output of heater is lower if it is whiter. You're trying to design the heater to be an "anti-insulator": you want the heat to leak out of the flow circuit in at the heater, not through the lagging on the hot water pipes outside the building channelling the water from the boiler to the heaters. If the hot water pipes leak heat in the same room, then that's no problem.
Recall the quartic dependence of the Stefan Boltzmann law. At room temperatures with a low temperature heater (the hot water kind) $\sigma\,T^4$ is likely to be pretty small compared with other heat transfer mechanisms, in contrast to my idealised scenarios above. So the heater's colour is likely to be pretty irrelevant.
I'm fairly certain that the energy difference between the two is negligible. Any heat as a result of light absorbed is localized on the roof, with layers of insulation between the inside of your house and the additional heat generated. Most of the additional energy absorbed would heat up the air rather than the inside of your house. Additionally, the optical properties between the roofs seem very similar; I don't think one would absorb significantly more light than the other, just at slightly different frequencies.
A much more significant effect on energy consumption are your windows. In the summer, these let in almost all the light that hits them, which is converted into heat, inside your house. To mitigate this, they make windows with reflective coatings In the winter, your windows act as good heat conductors, cooling down your house. You'll sometimes see double-paned windows. The layer of air between window panes acts as an insulator.
They also design houses where the windows let in light in the winter, and block light in the summer, based on the orientation of the window and height of the sun in different seasons.
Best Answer
There are two main ways that a house (or indeed any other object) exchanges heat with its surroundings: convection and radiation. As a general rule, at everyday temperatures convection is faster than radiation so it's the dominant mode of heat transfer.
With convection it doesn't matter what colour you paint your house. Convection heating and cooling is mostly by the wind blowing against the house walls and exchanging heat with the walls by conduction. In particular at night when it's cold the house will lose heat at the same rate whether it's painted white, black or yellow with green spots.
But the radiation from sunlight has a temperature of about 5,700°C so it is very good at transporting heat, as indeed you can tell just by standing in sunlight for a few minutes. Painting your house white (or better still silver) will reduce the absorptivity because it reflects away a large proportion of the sunlight, so it will reduce the rate of heating by the sunlight while not affecting convection.
So painting your house white will reduce the amount it heats up during the day but will not affect the amount it cools down at night. The end result is that it will keep the house cooler.
In winter the sunlight is often very weak or it's cloudy, in which case convection dominates and the colour of the paint has little effect on the internal temperature. It's true that on the rare sunny days in winter the white paint will reduce how fast the house heats up, but in hot climates this is a price worth paying for keeping the house cool in the summer. I suppose ideally you'd repainting the house twice a year so it was white during the summer and black during the winter.