Trouble is only in semantics, as far as I'm concerned, with "feeder" relationships being the worst - as financial and friendly agreements seems to be the most practical use of the feature.
As far as rivalry goes - I'm not familiar with the Scottish lower leagues, but I guess local deribies are matches of special importance over there, as it is in the rest of the footballing world. An example from my home town: When Fyllingen joined Brann in the Norwegian
prem a decade or two ago, rivalry and fierce derbies occured. Now, when Fyllingen is mid-table in div 2, nobody cares - I guess most Fyllingen fans support Brann in their efforts. If they were in the same league again though, rivals would be quite the appropriate term.