In cryptography, strong primes have been used (with various definitions of that) for RSA, in order to defeat the factorization of the public modulus by Pollard's p-1 and p+1 algorithms, and various other attacks. For this reason, there is no list of standard large strong primes, for that would defeat the purpose in RSA, where the prime factors of the public modulus must be secret.
The Diffie-Hellman key exchange often uses safe primes, which is a prime $p$ such that $\displaystyle q=\frac{p−1}2$ is also a prime. The answer linked to in the question accidentally misuses the word strong prime where safe prime is meant, as noted in comment.
Standard safe primes for use in DH are in RFC 3526. These intentionally use the generator $g=2$ for the subgroup of prime order $\displaystyle q=\frac{p−1}2$. Except perhaps for the 1536-bit one, they are considered large enough to be safe for use right now.
With (safe prime $p$), can I use any number between $2$ and $p−2$ and be guaranteed to have an order of either $p−1$ or $\displaystyle\frac{p−1}2$ ?
Yes. Argument: the order of a subgroup always divides the order of the group, which is $p-1$ for prime $p$. When further $p$ is a safe prime, the order of the group is $2\,q$ with $q$ prime, thus the order of any $g$ is one of $1$, $2$, $q$ or $2q$. Only $1$ has order $1$ and only $p-1$ has order $2$.
But as noted in comment, there is no good reason not to use the generator coming with the prime.
