"... even the slightest reference to those entities of which it can be said "absence of proof is not proof of absence"-- such as G-d ..."
Is that one of things (can't remember the name of that darn word) - like, is the sky is blue because it reflects the ocean, or is the ocean blue because it reflects the sky?
I'm not sure what word you were looking for there however, the ocean reflects the sky's apparent color and not the other way around, (water has reflective properties and the "sky" consists of a non-reflective atmosphere). The two are therefore not interchangeable or equivalent to the "absence of proof"/burden of proof dichotomy.
In the context which "elandry" posited that "it can be said", it can be but, not accurately. This is because a position which posits the existence of a "g-d", (whether directly making the claim or, such a claim being inherent in a belief in a supernatural entity), is implicitly making a claim for something which has no attributable evidence. That position then holds that the absence of evidence supporting the religious claim does
not constitute evidence that what is claimed is 'absent', (doesn't exist). The distinction between a 'positive' assertion and a 'negative' one is that one isn't required to produce evidence of
what isn't, (negative assertion), but is required to produce evidence of 'what is' asserted, (under the burden of proof requirements).