Preimage of a constructible set in spectrum of a subring

While working through a proof of this paper, at the beginning of page 42, the author seems to claim the following is true:

Let $ R\subset S$ be rings, where $ R$ is a finite type algebra over $ \mathbb F_p$ . Consider the associated map of the prime spectra $ $ \varphi:\text{Spec}(S)\rightarrow \text{Spec}(R). $ $ Suppose that $ K\subset \text{Spec}(R)$ is a constructible subset such that $ \varphi^{-1}(K)=\varnothing$ . Prove that there exists an $ R\subset R^{‘}\subset S$ , such that $ R’$ is a finite type $ R-$ algebra and that if $ $ \psi:\text{Spec}(R’)\rightarrow \text{Spec}(R). $ $ is the associated map of spectra, then $ \psi^{-1}(K)=\varnothing$ .

I believe that I have an argument for the case when $ K$ is a finite subset. One could think of $ S$ as a direct limit of its finitely generated $ R$ -subalgebras and therefore $ Spec(S)$ should equal an inverse limit of the spectra of the finitely generated $ R$ -subalgebras. For each prime in $ K$ , choose a finitely generated $ R$ -subalgebra where it does not have a preimage, and the rest is clear. However, I don’t know what to do for the general case.

preimage resistance

I’m struggling to get a clear understanding of second preimage resistance and collision resistance.

Research on the internet yielded the following definitions:

Second pre-image resistance

Given an input m1, it should be difficult to find a different input m2 such that hash(m1) = hash(m2). Functions that lack this property are vulnerable to second-preimage attacks.

Collision resistance

It should be difficult to find two different messages m1 and m2 such that hash(m1) = hash(m2). Such a pair is called a cryptographic hash collision. This property is sometimes referred to as strong collision resistance. It requires a hash value at least twice as long as that required for pre-image resistance; otherwise collisions may be found by a birthday attack.

As far as I understand, every collision resistant hash function is also second pre-image resistant.

I don’t understand why collision resistance is harder to achieve, given that input m1 of second pre-image resistance could still be theoretically any input in the domain of the hash function.