No, only tax-free employer assistance counts. An employer can pay up to 5,250 towards an employee's tuition and the employee does not need to claim it. Employee Assistance and Veteran's Assistance must be applied to tuition. Scholarships and Pell grants can be allocated in several different ways, against the tuition (making it tax-fee) or for room and board (making it taxable but freeing up expenses for a possible credit).
Box 1 should report what was paid to the school. It should not need to be altered, but if it does, that would be the way to do it.
Don't enter amounts twice, if an amount is already on the 1098-T, don't enter it again. If you have employer assistance on your W-2, don't enter that a second time either.
If the school reported the reduction in Box 5 and also subtracted it out of box 1, then they reported it wrong and yes, you would adjust for that.
IRS Pub 970 can be very helpful.
IRS Pub 970
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