Shouldn't Schoof's algorithm be taking generator point as an input ? All implementations I tried (including Sage) need only the coefficients of the Weierstrass equation and a prime. No mention of the generator point at all.
For elliptic curve y2 = x3 + 7 MOD 199, generator (11,12) gives a total of 63 points (including the point at infinity). While for same elliptic curve, generator (5, 27) gives a total of 21 points.