The reason different programs give different results is because they define what constitutes inside and outside of the circle differently.
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You could also use the distance from your [centre, of either kind] to the farthest corner, or the closest corner. Or the radius has to contain three corners, or just two corners.
Or go for tiles with more than half covered by the circle, in which case are you approximating a straight line between the two intersections or actually calculating the bulging arc. And though this would cover an absolute minority of cases, is it >0.5 or >=0.5?
Alternatively, anything completely within the circle is not the circle wall. Anything completely outside of the circle is not the circle wall. Anything lying on the circle (and possibly you need to fudge things if you get the circle intersecting a four-tile intersection, leaving a diagonally-navigable passage, as you would, frexample, from a radius five circle, based upon a four-square intersection, at the eight different points represented by the interchangable coordinates {+/-3,+/-5}) and from that you get a wall. Albeit it looks far better at large scales.
Yes, different systems will use slightly different methods. Use what looks good. Or an alternative (not exactly equivalent) philosophy is if it looks good, use it.