You have it right. But the situation is complicated by people not wanting to lose face. I started working for DEC in the mid '80s, and about 10 yrs later they were acquired by the relative newcomer Compaq. But all the internal FAQs we were fed had:
Q Is this a merger or an acquisition?
A It is a merger... [blah, blah, blah...]
If you've read '1984', you'll have met the concept 'doublethink. Somehow, the people who organized the acquisition found a way to bend the language so that the could justify the use of the word 'merger'. It made the few surviving DEC board-members feel better.
A few years later, HP acquired Compaq. Same story, even down to the logic-chopping internal FAQs.
The moral of my story is, hang on to your understanding of the difference between an acquisition and a merger, but treat with a bucketful of salt anything that the resulting company says about it.