Try changing the third include directive back to #include <mysql.h> and use the -I, -L and -lmysqlclient switches when compiling and linking
gcc -c -I/var/tmp/mysql/usr/include/mysql program.c
then
gcc -o program program.o -L/var/tmp/mysql/usr/lib/mysql -lmysqlclient
..or wherever the header files and client library are located