Two-wire DC lead, one white and one white with grey stripes — which is positive?

Do not rely on the marking. There is a loose convention that the marked or striped conductor is negative, and it is honoured often enough to be tempting and violated often enough to destroy equipment.

I have shortened leads from a single batch of identical supplies and found the polarity swapped between units. Whoever assembles the cable is not always the person who decided which side the stripe goes on, and there is no standard that binds them.

The reliable sources of truth, in order:

1. The symbol printed on the supply. Almost every adapter has a small diagram showing a circle with a dot and a C shape, indicating whether the centre pin is positive or negative. That tells you what the device expects.
2. Measure it. A basic multimeter costs very little and is the correct tool.
3. Check the device's own label, which usually repeats the polarity requirement.

The symbol plus a measurement is two independent confirmations, which is what you want before connecting anything you care about.

Since you have no meter to hand, there is a classic improvised test that genuinely works for low-voltage DC: use a potato, or a glass of salty water.

With the supply unplugged, cut and strip the two conductors and separate them well. Push the two bare ends into a potato a couple of centimetres apart, or into a glass of salted water without letting them touch. Then power the supply.

Electrolysis produces visible bubbling and, in a potato, a greenish discolouration. The bubbling is more vigorous at the negative conductor, and in a potato the corrosion mark appears at the positive one.

It is crude, it is slow, and it is a real technique that predates cheap meters. Keep it to genuinely low voltage DC, use small currents, and wash the potato off your hands afterwards.

Having said all that: buy the meter. You will use it for the rest of your life and it removes this entire category of question.

One caution about the project rather than the wire.

Check what the device actually needs before you commit — voltage, current, and whether it wants regulated supply. Network terminals and similar equipment are sometimes fussier than they look, and some expect a supply that can deliver a surge well above its steady draw.

Also check whether the battery backup you are planning to use outputs a waveform the original adapter is happy with, if you are keeping the adapter in the chain. Cheap backup units produce an approximated waveform that some supplies dislike, and the failure shows up as an intermittent reboot months later rather than immediately.

Practical tip once you have identified the conductors: mark them permanently before you cut anything else.

A dot of paint, a piece of heatshrink, a label. The number of times I have carefully determined a polarity, put the job down, and come back the next day to two identical white wires is more than I would like to admit.