Postgraduate programs are, as you've discovered, structured very very differently in the USA and Europe.
First, the caveat: there is doubtless lots of variation across institutions and fields, and I'm basing the below on my limited experience (PhD from UC San Diego and now a postdoc at U. Edinburgh, so I know lots of current UK PhD students). Also, I'm in exactly the same murkily defined field that you're asking about.
My impression is that the general rule is: in the US, PhD programs and research-oriented masters programs are treated together as being a single indivisible unit. This has obvious practical consequences -- for example (and in answer to one of your questions), in the US PhD programs I'm familiar with, if you show up with a masters in hand, this doesn't actually change your course requirements versus someone with only a BA/BS. You'll have to go through a second "masters program" (but probably won't officially be awarded a second masters degree). But it also affects the structure of the program itself. My program had required lab rotations and a structured "2nd year project" that were technically part of our masters coursework, but were explicitly designed to provide a gentle ramp into our PhD research. (Psych departments are usually even more aggressive about this, and will have you running subjects within the first 3 months, ideally in the lab you'll continue in through your PhD research.) You're working in the same environment, with the same cohort, and the same faculty throughout, and the expectation from the beginning is that you'll be going all the way through to the PhD, so there's training from the start on how to "think like a researcher" (which is the main thing taught in PhD programs), the programs can kind of blur into each other, and there's some flexibility about when exactly you start your PhD project proper. This potentially gives more room to try out things that don't work, etc., before you have to write your actual proposal and put your nose to the grindstone. OTOH, it's not uncommon for programs to stretch into 6 or 7 years.
In the European system, by contrast, masters and PhD programs are treated as distinct programs. Sometimes people stay with the same supervisor for both, but it isn't the usual case, and the programs aren't structured with that in mind. The PhD programs as far as I can tell are much more tightly defined: you're expected to show up, start work on a project more-or-less immediately (possibly a specific one that your supervisor has picked out ahead of time and gotten funding for), and be done and dusted in 3 years.
For me, the US system was definitely better, and that bias is probably reflected in the above; but, people are different, and I can imagine the European system working better for others. Also, every PhD is different -- by far the most important determinant of PhD success is the interaction between you and your supervisor. (Well, being funded is pretty important too.)