Sorry that your advisor is becoming a bear about your graduating from MA.
A department makes the recommendation for admission. It is the responsibility of the Graduate Studies office to check transcripts, eligibility etc. and yes, also making a conditional admission. The fact that your Dept. Chair did not reference the conditionality of admission, does not mean it is not conditional. It is still conditional that you finish your MA. Even if your advisor had not written about it in his LOR, if your transcripts showed that you were still finishing up your MA at the end of Spring Semester, it would trigger the conditional clause for admission. (I worked at Grad. school office and have seen this happen several times)
If you do not tell your PHD program - that could come to bite you. They can revoke your admission and take back any money they have paid you. Are you likely to finish the thesis during Summer and only need to return to defend it in the Fall or do you think you will have to continue working on it in the Fall?
Check your PhD. school policy on deferring your admission to next Spring. Also check the policy on conditional admission. Some schools allow you to submit updated transcripts to show completed coursework before you go for your comps.
Good Luck!