I think it was a huge mistake for Democrats to also adopt and push the name "Obamacare". I have thought this from day one. It opened up the possibility for seeds of confusion to be sewn by the enemy.
Republicans first coined the term because they thought it would eventually fail, partly because they were going to do their utmost to make it fail. And they wanted to tag it with his name so he went down with the ship. And also it removed the nice sounding "Affordable" from the equation.
Democrats started then using it too. I heard Obama himself use it on many occasions. The reason they did was an act of hubris and arrogance. They were so sure that it would catch on and be popular that they thought they'd call the GOPs bluff. Thinking that one day the GOP would rue the day they attached Obama,s, a Democrat, name to a popular game changing program.
They should have insisted on calling the Affordable Care Act whenever and wherever they were asked to talk about it.