Am I right in thinking that these papers need a green safe light and not the more common red?
No, that makes no sense. Apart from (today, no longer manufactured) panchromatic papers, no papers are sensitive to red light. Red sensitivity is something an emulsion needs to be sensitized for specifically and there has never been a need for this in printing from B&W negatives. Hence, a red safelight is always a good choice. This is under the condition that the safelight is truly red, and not also emits light in other parts of the spectrum. Red leds are often problematic in this sense as they tend to have secondary emission peaks in the green part of the spectrum, which necessitates additional filtering.
But an old-fashioned fixed grade paper that works OK under a yellow safelight will also work OK under a red safelight.
If you observed fogging with your red safelight, it's either due to the paper and its age itself, or it's due to the light not being pure red.