Question

1. Jack must have been late for the school. He got up late.

2. Jack must have been late for school. He got up late.
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What is the difference between 'school' and 'the school' in the sentences?

Does #1 mean he was late 'for another school,' not 'for his school'?
Does #2 mean that he was late for his school?

Answer 1

No one would ever say #1.

Answer 2

When you write or say "the school," you are referring to a particular building or group of buildings. Steve is right; it does not mean the same as 2, and would not be used.

You are right about 2.