Is a hash the same as encryption?

No. Encryption can be reversed with the key. A hash cannot be turned back into the text. The same text always produces the same hash, and a different text produces a different one.

Which hash should I copy?

Copy SHA-256 unless the checksum you were given says MD5, SHA-1, or another name. A 32-character value is MD5. A 64-character value is SHA-256.

Why does my hash disagree with the command line?

The command often hashes a trailing newline that you did not type. A space at the end of the text changes the hash too. Hash the exact characters, or hash the file itself.

Can I store a password as an MD5 or SHA-256 hash?

No. Both are too fast for a password, and MD5 collisions can be built on purpose. Use a password hash made for that job, and use a long random password in the meantime.