past tense of must in reported speech

1.) "You must be wrong," he said. (probability)

will be:

He said I must be wrong.

and not

He said I had to be wrong.

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2.) "You must appologize," he said. (external moral obligation)

will be:

He said I had to appologize.

and not

He said I must appologize.

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Am I right in thinking that 'must' goes through a modal change if and only if it expresses obligation or necessity?

Please, note that I made up the sentences myself -- they might be incorrect.

Thanks.