Please see the disclaimer.
tl;dr: The Church did not change its name. It reëmphasized the actual name of the Church, one given by revelation.
First of all, the title of the article, “Why the ‘Mormon’ church changed its name”, is wrong. The name of the Church was not changed.
The name of the Church has been changed only a few times, all in its early history. Only one of those, the last change, came by revelation, as recorded in D&C 115:4. That was in 1838, only eight years after the restoration (founding) of the Church.
Since then, for 180 years, the name of the Church has remained the same (besides inserting a hyphen and changing capitalization in 1851), so no, the Church did not change its name.
This is one thing that the article gets right: it's not a rebranding; it's a reëmphasis. It is like the time in 1995 when the logo of the Church was changed.
The image above shows the change. On the left is pre-1995. On the right is after.
The Church changed the logo to make the name of the Savior bigger, since he is the real Head of the Church, not a man with the title “President.” That reëmphasis is the same as this one: we are making sure that the world knows that we follow Christ before anyone else, not Russell M. Nelson and not Mormon, the ancient Prophet who compiled The Book of Mormon: Another Testament of Jesus Christ.
Second, I would like to talk about this quote from the article:
When a president calls his decision “God's will,” that essentially ends any argument.
This sentence was written by someone who does not understand how God speaks to us.
It is true that the Prophet may make decisions of his own accord; that is part of his calling as the president of the Church. He may ask for revelation, and God may not give him any, trusting him to make decisions on his own. When such decisions happen, he will not say that it is from the Lord; he will simply give the direction.
In that case, we know that the Prophet has made a decision on his own. However, we have a promise from a prophet that the Lord would remove him from his place before allowing him to lead us astray.
But none of that applies when the direction is from the Lord, which is what happened here.
When the Prophet says, in any way, that he is speaking for the Lord, we can assume he is because of what I said above: the Lord would not let the Prophet lead us astray and would remove him out of his place if he did so.
On top of that, we have a promise: that any direction that the Lord gives to the Prophet will be confirmed to us, if we but ask Him for that confirmation.
And if we have it confirmed, we are no longer blindly following a man; we are following God with perfect knowledge.
So why is the quote above wrong? Because the Prophet never calls one of his own decisions “God's will.” If he says that a decision is God's will, then he did not make the decision.