just a thought:
order_id = '944704' OR ticket_id = '944704'
order_id = 944704 OR ticket_id = 944704
would do the same, mybe mysql is doing some conversion there
also I would try
`visible` char(1) NOT NULL DEFAULT 'Y',
`visible` INT NOT NULL DEFAULT 1//as 1 = 'Y', 0 = 'N'
EDIT: since you mentioned database class, do you have connection pooling ?
If not, opening connection each time can eat substantial amount of that x second of yours.