Here is a related question when-is-it-appropriate-to-select-models-by-minimising-the-aic?. It gives you a general idea of what people not unrecognizable in academic world consider appropriate to write and what references to leave in as important.
Generally, it is the differences between the likelihoods or AICs that matter, not their absolute values. You have missed the important word "difference" in your "BIC: 0-2 is weak" in the question - check Raftery's TABLE 6 - and it's strange that no-one wants to correct that.
I myself have been taught to look for MAICE (Minimum AIC Estimate - as Akaike called it). So what? Here is what one famous person wrote to an unknown lady:
Dear Miss --
I have read about sixteen pages of your manuscript ... I suffered exactly the same
treatment at the hands of my teachers who disliked me for my independence and passed
over me when they wanted assistants ... keep your manuscript for your sons and
daughters, in order that they may derive consolation from it and not give a damn for
what their teachers tell them or think of them. ... There is too much education
My teachers never heard of papers with titles like "A test whether two AIC's differ significantly" and I can't even remember they ever called AIC a statistic, that would have a sampling distribution and other properties. I was taught AIC is a criterion to be minimized, if possible in some automatic fashion.
Yet another important issue, which I think have been expressed here a few years ago by IrishStat (from memory so apologies if I am wrong as I failed to find that answer) is that AIC, BIC and other criteria have been derived for different purposes and under different conditions (assumptions) so you often can't use them interchangeably if your purpose is forecasting, say. You can't just prefer something inappropriate.
My sources show that I used a quote to Burnham and Anderson (2002, p.70) to write that delta (AIC differences) within 0-2 has a substantial support; delta within 4-7 considerably less support and delta greater than 10 essentially no support. Also, I wrote that "the authors also discussed conditions under which these guidelines may be useful". The book is cited in the answer by Stat, which I upvoted as most relevant.