I see no proof, nor anything that even hints “that you need both relevant and irrelevant backlinks to rank #1”. What makes you think that irrelevant backlinks would ever be needed to rank for a keyword? That doesn’t even make sense, much less is there any proof of it in your examples.
Those terms you posted as examples have very little value in ranking since according to Google they both get fewer than 10 searches per month. So most webmasters could rank for those keywords with very little effort, perhaps even by accident. Probably not the best choice for demonstrating your SEO prowess.
I believe the whole notion that one needs “irrelevant” backlinks to rank is just pure Cargo Cult SEO, with absolutely no basis in truth.