Well, I think that on the Trump side of things they do not want what happened on January 6th to be called an insurrection because that makes Trump guilty of insurrection and this means he should not be allowed to hold public office again. So, here is the definition of insurrection:

An organized attempt by a group of people to defeat their government and take control of their country, usually by violence.

Basically, if we just look at the definition of the word, January 6th qualifies as an insurrection. January 6th was organized for over a month and the event was being promoted as "Stop the Steal" which specifically meant overturn an election won by Joe Biden so Trump could defeat the government and take control of the country. Violence was used to achieve the goal. 

Trump's lawyer in the 14th Amendment case that was just before the Supreme Court called January 6th a "violent criminal riot" which implied there was no planning involved, Trump's supporters just suddenly got out of control. This was not at all true. Many meetings were held prior to January 6th, tours were given of the Capitol to make sure the insurrectionists would know where to go, and Trump and his lawyers and representatives held several meetings to discuss how to disrupt and stop the certification of the election. The day was marked as the final chance to prevent Biden from being able to take office and prevent the peaceful transition of power. 

So, by the definition of the word "insurrection" January 6th most certainly meets the definition...to a T.

It was preplanned, designed to defeat and take control of the government and it was violent. There was more than one militia group there that had met with Trump's lawyers and reps and planned for weeks how to get into the Capitol, stop the procedure of reading and certifying the electoral count, and to, if possible, kill people like Mike Pence and Nancy Pelosi. Trump was aware of what was happening and took part in the planning and then ordering the crowd to march down to the Capitol to make sure if Mike Pence did not do what Trump wanted, they would.