What does it mean when a charger has 5V 0.7A?
A power supply specifying 5.0 V and 0.7 A output means that the voltage across its output should be 5.0 V while any current between 0 A and 0.7 A is drawn by the load on its output.

Your measurements would indicate that none of your power supplies are meeting that specification exactly; but there are two effects you must consider:


Tolerance. No voltage regulation can be perfect, and there is no reason for a typical power supply to be that precise. USB 2.0 specifies a supply voltage of 5.00 ± 0.25 volts, so a minimum of 4.75 volts.
Voltage drop due to the resistance of the cable attached to the power supply. (I assume the figure for your lab power supply came from the meter of the power supply itself, thus showing approximately no drop.) This results in a low reading (and a lower voltage supplied to the device) as more current is drawn. It is possible for the power supply to compensate for a known cable resistance, but this is not usually done.


So, your measurements (assuming they were made correctly) are quite consistent with the phone refusing to charge (or charging at a low rate) due to the input voltage being below 4.75 volts. The power supplies might meet their specification in their internal circuit, but if so, the permanently attached cable contains conductors which are too thin, and so the whole must be considered out of specification.

(Note, however, that there is no obligation on the phone manufacturer to set a limit of exactly 4.75 volts. In fact, unless you connected the proper resistors to the data lines when you tested using your lab power supply, we can see that the phone does not follow USB specifications for allowable current draw. This is quite common.)