Orignal Schnorr Signature vs Discrete logarithmic Schnorr Signature
The first publication using the Discrete Logarithm Problem (DLP) for asymmetric cryptography is in Diffie-Hellman key exchange. The original works in the multiplicative group $\Bbb Z_p^*$, or a typically large subgroup of that.

The first use for signature is ElGamal signature. It uses the same kind of groups, and that makes its signature rather large. It does not use a hash. That and other issues make the original description unsatisfactory; it's more a building block for later signature schemes based on the DLP.

The most notable is the original Schnorr signature. It works in a slightly different group, called Schnorr group, which is a small subgroup of $\Bbb Z_p^*$, with $q$ elements where $q$ is a prime dividing $p-1$, and much less wide/large. That provides what long was (and if we restrict to practice, remains) the shortest known secure signature with appendix (that is, added to the unmodified message): $3b$-bit signature for $b$-bit security. The signature consists of a $b$-bit hash and an integer modulo $q$ over $2b$ bits.

A variant differing only by using a $2b$-bit hash is cleaner from a math standpoint and slightly less difficult to prove secure, thus was much studied, and is stated as being Schnorr signature in many textbooks. That one has a $4b$-bit signature, for a group with a little less than $2^{2b}$ elements. The exposition in the HAC is among these, and it's signature has the hash second, contrary to the original. It is followed by an example with small integers.

The later DSA of FIPS 186 is often seen as a way to work around Schnorr's patent. It is significantly different and more complex. It has $4b$-bit signature.

There are many later signatures also called Schnorr signature, typically with $4b$-bit signature, also based on the DLP, but in an Elliptic-curve group. Study these later!




"original Schnorr signature" unambiguously means a scheme in a Schnorr group, with $3b$-bit signature having one component a $b$-bit hash, the other $2b$-bit coding an integer.
"Schnorr signature" can be any number of things. In introductory material that could be either the original, or the variant where the hash is made $2b$-bit. Other possibilities are too many to enumerate.


Note: "Discrete logarithmic Schnorr Signature" is a naming error. "Logarithmic" is an adjective applying to quantities, mostly continuous. A complexity can be logarithmic, perhaps by extension a count including the bit length of a bitsring, rather not a bitstring like signatures are, not a signature scheme which is a method to obtain and verify a signature. Make that: "Schnorr signature" (thus) based on the Discrete Logarithm Problem (in some group).

Remark: some videos are a great way to learn for their author.