I am a PhD student at a European university, and national of an ESTA country. I will visit the US for 2 months as I informally collaborate with a professor based at a US university. I plan to conduct my own research there profiting from his advice and the opportunity to meet him more regularly, while occasionally visiting lectures or conferences there. During my stay, I will be invited to give talks at other universities and meet with other professors, or visit conferences. I won't be paid by any US university nor work for any US institution, and I have no official affiliation with the US universities or professors I will visit. As such, I have no official invitation letter from the university or anything. I also expect to tour the US a little bit, especially since I have a friend based where that US university is.
My understanding is that I could go to the US simply with an ESTA, as I'm not working there nor receiving US income, but conducting independent research. If so, should I mention business as my primary reason for visiting or tourism? Or should I get a visa of some sorts instead?