This is almost certainly down to the lack of a proper connection between the signal ground of the amp and the power supply ground. The external PSU is very likely a 'double insulated' PSU (there should be a square in square symbol on it if it is).
It's electrically safe, but the noise picked up by the cable screen has nowhere to go. Same thing will happen when it's running on batteries.
It's becoming more and more of an issue with laptop-based recording systems and bus-powered interfaces, where there is no linking between the system ground and earth, or else it's a fairly high resistance path.
You can normally cure this by linking an exposed metal part of the amp or cable to earth. You can buy boxes to do this if you want, but a simple DIY small metal FX enclosure, with an in and out jack (metal chassis guitar-style jacks, not the plastic type), linked straight through, with a long earth wire attached to the metal case with a mains plug on the end and the wire connected to the ground pin.
Your Fender amp will have a properly earthed chassis with the signal ground connected to earth, due to the internal power supply.
Gold-plated jack plugs aren't as good as standard jack plugs in terms of conductivity. It's got about 1/4 of the conductivity of chrome IIRC (though obviously it's a very thin layer so overall resistance is barely affected). The benefits of gold-plated connectors are for connectors that get put together once and left e.g. in a hi-fi system, as they won't corrode. Normal style guitar jacks get cleaned of corrosion every time you insert or remove the jack plug, which also scrapes the gold off quite quickly. So gold connectors are best for for fixed cables, chrome connectors for regularly used ones.







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