Washing machine using cold water only

Washing machines don't just use hot only or cold only; they often mix the two. When this happens, the pressure inside the valves will push some of the water out the hot connection.

How to fix this? Buy a short, cheap, flexible hose, cut it in half, fold over the end, and hold it down with a zip tie. Then screw the end into the washing machine. Should be close enough for government work.

The hot and cold water enters what amounts to a mixing manifold inside the washing machine. When cold water flows into the manifold, the pressure inside the manifold is necessarily higher than ambient pressure. Since both the hot water electromagnetic valve as well as the feed hose are now open, they allow water to flow out of the manifold. You should remove the hose from the washing machine and plug the hose attachment nipple using a flat screw-on plug cap with a sealing washer inside. It's possibly cheaper than butchering a hose to fashion a plug.

The same thing would happen on a faucet

if you did the same thing -- leave the hot pipe dangling then set the faucet to "Warm".

Any position other than 100% hot or 100% cold involves mixing both hot and cold. That means hot and cold are connected both to the washer inlet at the same time. Water follows all paths in proportion to its conductance (1/resistance). It's easy to go backwards out the connected "hot" inlet. So it does. Washing machines don't have check valves to prevent this, why would they need that?

Stop using "Warm" settings on the washer.

And done.

Use only "Cold" setting, so it isn't opening both cold and hot at once.

"Hot" would also avoid opening both valves at once, but the washer won't fill if you do.

When you think about it, asking the washer for "Warm" when you know hot isn't even connected doesn't make much sense.