Apple - How to reset a magsafe charger that doesn't charge macbook pro's battery?

Sounds like you got a sticking ground pin.

Just plug it out when connecting to the laptop horizontally (with more force) and stick it back it horizontally as well. Usually this is due to you removing the plug by lifting it (feels gentle but actually bad) rather than yanking it.

A MagSafe Power adapter cannot be reset. It's not a function of the computer but a failure of the internal component(s) of the power supply itself.

Why does it fail? IBM engineers addressed that very point in a presentation - Why do Power Supplies Fail, and What can be done about it?

Reasons range from environmental factors, quality/age of the components, and the spacing/location of components (design).

You've already done your diagnostics (trying a different one) and you have narrowed down the issue to the MagSafe adapter itself. It's time to get a new one.