secp256k1 prime modulus vs order For curve secp256k1 prime modulus is $2^{256}-2^{32}-977$ and order is smaller number but has near half of starting bits set. If I draw number to be private key, it must be less than order. All field operations are modulo prime modulus and it means they must be smaller than modulus but can be  &gt;= order? (or not?). Coordinates of public key point can exceeds order?
A very interesting case is in signing and verifying: in pseudocode: from nayuki library is modulo order.
 * if (nonce outside range [1, order-1]) return false
 * p = nonce * G
 * r = p.x % order
 * if (r == 0) return false
 * s = nonce^-1 * (msgHash + r * privateKey) % order
 * if (s == 0) return false
 * s = min(s, order - s)

This means, operations are done, first modulo modulus p, next modulo order?
Yes, field operations may involve numbers that are larger than the order. Public key point coordinates may exceed the order.
Parts of the protocol that handle scalars work modulo the curve order, while the parts that handle elliptic curve points work modulo the prime modulus.
Sometimes, as you noted, some information &quot;crosses&quot; between these domains: the main case being inside the ECDSA protocol, in the r value which is obtained from a X coordinate and reduced modulo the order to enter in the computation of s.