20th Century Fox paid for the Special Editions, yes. It was on their dime, and I believe it was initially their idea, too. Lucas ended up negotiating with them further, and used the money they’d initially earmarked for a 20th anniversary restoration and re-release of Star Wars to be a “Special Edition” trilogy project (I believe they got Fox to increase the budget at that point as well) but the story of the Special Edition starts with 20th Century Fox approaching Lucas about getting the original film back in theaters as an event, and Lucas then using his leverage to fix things he wanted to fix then, and add things he couldn’t have added, especially since Fox agreed to pay for it.
Now the quality of the “fixes” can be debated, and have been for over 20 years now, I’m not going to argue that the execution was successful. But the intent of the Special Editions was definitely different than the intent behind the DVD release, and the later blu-ray release. The Special Editions were initially begun with a bigger idea of “fixing” the movies so that they looked and felt more like he wanted them to back then. The DVD and Blu-Ray additions - not paid for by Fox - were undertaken more along the idea of “this seems cool, put it in there.”
Director tinkering is the best description for all three edit passes. It was the same thought process and the same type of changes. Only TESB had some real technical fixes (recompositing the snow effects shots). The rest were outright changes to the entire frame or recompositing new effects over existing footage. There really isn’t much different between changing the door in the 2011 version of ROTJ and changing the sandcrawler in the 1997 SE. Who was paying for it is irrelevant to the nature of the changes. It was all GL tinkering with his films to “improve them”.