Oh yes. Its nothing illegal as far as i know.
It has your College name and in the remarks section it says that AUTHORIZED TO UNDERTAKE EMPLOYMENT WHICH FORMS INTEGRAL PART OF STUDIES..
I also worked on that work permit, on full time basis after i finished my courses in the class room. So 3 months of internship on that work permit. And once those 3 months were done, i was employed full time by the firm and continued to work on that WP till i got my PGWP. So its nothing illegal about working for an employer other than your college (even though college name is printed as EMPLOYER on WP). If you are still worried, just email that initial college where you did that english course, as a NEW prospect student, and ask them if you can work when you will come to canada on this English program. Ask them if it would be legal for you to work while studying - and they will elaborate.
And if you are still not sure - (by the way ..then you will be in paranoid group

)...you can always double check with a lawyer. Its good to spend $250 in the beginning than loose $550. Though, there is no need for this option. Its your call.
All the best.