A string in C is, by definition, null terminated.
Note the second function is off by one in the opposite direction-- it doesn't find "ok" in "ok".
strncpy, despite what its name implies, does indeed not necessarily produce strings.
Regardless, a C string is defined to have a null terminator. All bets are off if you don't provide a string.
Even with a null terminator, there isn't a length that would work for both without overrunning the buffer.
printf("%d %d\n", search_ok_1("mok", 2), search_ok_2("mok", 2));
printf("%d %d\n", search_ok_1("mok", 3), search_ok_2("mok", 3)); /* Overruns without null term */
printf("%d %d\n", search_ok_1("mok", 4), search_ok_2("mok", 4)); /* Overruns even WITH null term */
1 2
1 3
1 1