Thesql cannot start because of init.d folder

The problem is the mysql client program configuration. You can specify the --socket=/NEW_DIR/mysql/mysql.sock command-line option or add it under the [mysql] section of your my.cnf file.

Please look at the documentation for specifying options through option files: http://dev.mysql.com/doc/refman/5.0/en/option-files.html

Since changing the socket path affects all the client programs, then it will make sense to update the global my.cnf and set the socket option under the [client] section.

The two answers given from @RickJames and @drogart are essentially the remedies. (+1 for each).

Right from the error log you present, the last two lines say:

InnoDB: Error: log file ./ib_logfile0 is of different size 0 134217728 bytes

InnoDB: than specified in the .cnf file 0 268435456 bytes! `

At that point, it was evident that you set innodb_log_file_size to 256M (268435456) in my.cnf while the InnoDB Transaction Logs (ib_logfile0,ib_logfile1) were respectively 128M (134217728) each. Looking back at the link to my StackOverflow answer in your question, you had to do the following:

Step 01) Add this to my.cnf:

[mysqld]
innodb_buffer_pool_size=4G
innodb_log_file_size=1G

Step 02) Run these command in the OS

mysql -u... -p... -e"SET GLOBAL innodb_fast_shutdown = 1"
service mysql stop
rm -f /var/lib/mysql/ib_logfile*
service mysql start

So as to have confidence in what is happening, run tail -f against the error log. You will see message telling you when each innodb log file is being created.