Correct, the truncated versions of SHA2 are secure against length extension.
This prevents length extension because you've thrown away 256 bits the attacker needs to perform their attack.
On CPUs with hardware SHA256 (AMD Zen, 64-bit ARM, some recent Intel CPUs), SHA256 is faster.
For 64-bit CPUs without hardware SHA256, SHA512 & SHA384 are the fastest and they have identical speed (as only the values of some constants differ, while the algorithm is the same).
Most libraries and hash utilities implement all these variants, so any variant can be chosen without problems.