Remember that, in circuit analysis, "wires" are simply a tool we use to show connected components. You should try to see circuits as a combination of nets; a net simply describes how component pins are connected to each other. I know there can only be two different voltage readings in your circuit because your circuit only has two nets: ground, and 10V. Think about it. Every pin of every component in this circuit (in this case, a voltage source and resistor) is hooked up to only those two nets. No other nets exist.
Remember, voltage is always measured between nets, so there's really only one voltage to measure in your circuit, since you only have two nets -- ground, and 10V. It makes no sense to talk about the voltage at different points along a wire; just imagine the wire didn't exist, and the component(s) were hooked directly to each other. For example, if you hooked the little ground symbol straight up to your resistor (without the wire labeled "B"), that wouldn't affect the operation of the circuit at all.
Explanation: voltage doesn't change in an ideal wire. In other words, if you were to build that circuit, and measure the voltage (relative to ground) at every point along the circuit, you'd measure 10V all along the wire that hooked your positive voltage source to the resistor. Then, on the "bottom side" of the resistor, you'd measure 0V, all along the wire that hooked the resistor to ground.
Voltage only gets dropped across loads. Resistors, incandescent bulbs, motors, LEDs (and other semiconductor devices), etc.
So, if you added a second 1k resistor in series below the one you already have, you'd (again) measure 10V at the top of the first resistor, then you'd measure 5V in between the two resistors, and then you'd measure 0V at the bottom of the second resistor.
NB: We're talking about ideal wires here. In reality, every wire is a combination inductor/capacitor/resistor. It drops voltage. It stops sudden changes in current. It stops sudden changes in voltage. Wires are nasty things!
Also, you may have gotten confused by the "edge cases" that violate circuit theory; For example, if you replaced the resistor in your circuit with a wire, it would appear that the wire would have to drop 10V. But that circuit is invalid, and you can't apply analysis techniques to it.
The reason not to measure resistance on a powered circuit has more to do with possibly damaging the multimeter because of voltages that are applied from the outside. Because of the way ohmmeters work (they must pass a current through the circuit being measured, so their internal resistance can't be arbitrarily high) it's more difficult to protect them from externally applied voltage.
If the wall sockets are wired properly there's no risk of any damage doing this, but there is an outside chance you might not get an accurate reading because there might be a small voltage present between the two grounds due to some leakage somewhere in your house electrical system.
Why would you not just measure between the ground connection and the strap?
Best Answer
To first approximation, you can think of a battery as a fixed voltage source with a resistance in series. Your overall circuit therefore looks like this:
R1 and R2 form a voltage divider so that the voltage you see coming out of the battery is less than the battery's internal voltage.
For good batteries properly suited to the application, R1 is "low" compared to the effective load resistance (R2). The voltage drop is therefore usually small.
In your second example above, R1 is 9 Ω That's significant when the load is 40 Ω. In fact, with those two values, the voltage divider will give you (40 Ω)/(9 Ω + 40 Ω) = 82% of the battery's actual internal voltage.
If you just connect a voltmeter to the battery, then R2 is very large, like 10 MΩ, so you won't notice the tiny voltage drop due to R1 being in series. In fact, this is how you measure the battery's internal voltage (determine V1 in the schematic above).
Batteries are complicated, and in reality things are not this simple. The biggest gotcha is that V1 also varies with a number of conditions, like temperature, discharge state of the battery, immediate previous history, etc. But, the above is a useful enough first approximation to explain what you saw.