@brainwane and interesting question!
i did read the responses before i looked at the poll which may have biased me somewhat into just confirming my own preconceived notions but i would argue one of the key transition points into circle-ness is when you stop just knowing how many sides there are at a glance and have to actually count them—for me the polygons having a distinct visual identity starts dropping off after eight and is down to “it looks like an odd number?” by 13
i’d also be interested to know whether people would give different answers if, instead of being a regular polygon it was constructed “pixel art”
style with entirely 90° angles and just increasing numbers of “steps”