I like to think about the essence of the fallacies and what the ‘founding fathers of logic’ were thinking when they formed them. In doing so, I find two fallacies that could help us identify the error in reasoning in your example: argumentum ad populum (argument to the people) and argumentum ad numerum (argument to the number). Sometimes these are used interchangeably, but they have slightly different meanings.
What is a citation, after all, but a number of people who have cited a particular paper? And what is that but a measure of its popularity? Thus, using citations as a metric of an arguer’s power can be thought of as an error in reasoning in the same way that using the popularity of an argument is an error.
The ad numerum fallacy also invites us to realize that the size of numbers can lead us astray. Large numbers of discrete facts are meaningful when reasoning, but large numbers of echoes or affirmations of one fact are much less so. Put even more clearly, “more people have repeated my facts than your facts,” the essence of a citation, strikes me as a clear fallacy in reasoning. It seeks to win the argument by intimidation with numbers.
By the way, it's also easy to defeat this argument. Citations are merely a measure of what's popular in science at the moment. As in everything today, the more salacious a finding ("power posing causes huge increases in confidence!"), the more it will be cited. Click-bait doesn't just affect regular people. It affects scientist, too. Scientists have all the other human biases as well, including agendas, egos, confirmation and selection bias, etc. The problem is that many of the most popular findings in science aren't replicating. Indeed, science is undergoing a huge replication crisis right now because, apparently for the first time, scientists overcame their cognitive biases and actually started doing the boring work of attempting to repeat findings (scary, I know, since this is the very essence of the scientific method).
What happened? "According to a 2016 poll of 1,500 scientists reported that 70% of them had failed to reproduce at least one other scientist's experiment (50% had failed to reproduce one of their own experiments)." (Source: Wikipedia, https://en.wikipedia.org/wiki/Replication_crisis
Psychology (Dr Bo's field) is one of the hardest hit. In a look at "the reproducibility of 100 studies in psychological science from three high-ranking psychology journals," it was found "only 36% of the replications yielded significant findings... compared to 97% of the original studies that had significant effects." Oops.
This leads me to the fallacy you raised, the argumentum ad verecundiam. That’s a different intimidation tactic. It literally means "argument to modesty." The Latin root gave us the archaic English word "verecund," meaning "bashful, modest." I infer this connects with the logical fallacy because the victim of an ad verecundiam attack is supposed to submit, feeling bashful, when the implied authority is brought to bear.
To use an abusive form for emphasis, it would be something like: Are you a climatologist with an expertise in the Amazon rainforest? No? Then you don’t know what the hell you’re talking about. STFU.” Meanwhile, that has nothing to do with your argument, which may actually be correct in its facts and reasoning. That doesn't apply to your example (you were debating competing authorities), but you get my point.
For this reason, I reject arguments such as "be careful you don't confuse appeal to authority with deferring to authority.” I reject this as an intimidation tactic. Authority isn't all it's cracked up to be (see above). So many thing authorities told us in the past, which also had the weight of ad populum/numerum reasoning, were dead wrong (e.g. saturated fat is bad, you should replace meat with healthy low-fat foods). It happens so often, my friend came up with an acronym mnemonic: WDKS. We don’t know shit. Take authority with a grain of salt. Too much “authority” turns out to be false authority these days. Reason for yourself from first principles and trust the test of time.
And: Be especially suspicious of “authority” cited during exchanges on discussion forums. That’s just motivated reasoning with citation. ;-)