No, I am also talking about the same thing. But you correctly distinguish Europe vs. US. To make progress towards your PhD while working for company in the US,

No, I am also talking about the same thing. But you correctly distinguish Europe vs. US. To make progress towards your PhD while working for company in the US, you typically need to be enrolled in a department approved "class" that is basically an internship. To be enrolled in these classes, you need to be a full-time student. Most departments outside of CS and EE don't allow for these courses. Obviously, this is going to be highly nuanced with regard to how each university/department implements it. For example, my department only allows summer internships and requires approval from Advisor and Director of Graduate Studies.