It is a general piece of advice that for a successful PhD application one must demonstrate sufficient aptitude for and experience of doing research, and an ideal way for an applicant to do this is by participating in research projects as an undergraduate and/ or during a Master's degree.
PhD admissions committees will also be aware that many applicants are applying before finishing their Master's degree (if we are talking about UK PhDs here, where candidates are generally expected to have a Master's degree by the time they start- unlike the US, where someone going from Bachelors direct to a PhD is more common). This means that having a research project or thesis still in progress when you apply is not going to count against you.
"In the first university, and I will have done more semester projects throughout the year, but the Masters thesis is only at the very end of the degree, hence after PhD applications."
While you may be writing the thesis itself after the applications are finished, I think you will be aware of who your supervisor is going to be and what topic you will be working on well beforehand. If you choose this option, you will be able to discuss the future project in your research statement and interviews.
"The second option would consist of a Masters project in either the second or third semesters (thus, either before or during applications), although likely not in the areas I'm interested in."
This option doesn't sound so good to me. I made a point of finding a topic for my Master's thesis that I was very interested in, and it led almost directly to me getting a PhD place (even though I was barely halfway through the project when applying for the PhD). Admissions committees want to see motivation for your studies, and if you're not keen on your thesis topic this might be difficult to convey.
"...how many solid research experiences (equivalent to a Masters
thesis-level) would be needed for admissions into the top 10-20 US
From what I have learned by participating in this site, many US PhDs don't even require one Master's thesis- and I think, unless you're changing field significantly, it would be unreasonable to expect applicants to have any more than one Master's degree.
Finally, you mentioned in a comment that EPFL (I think) offer some neuroscience courses during the physics MSc. Given your goal of a neuroscience PhD, the opportunity to gain some experience in neuroscience before you apply can only be a bonus- one, because it will help you justify your motivation for applying; two, because you will be able to confirm for yourself that you really are interested in the field; and three, it may give you some ideas of topics you could focus on in your PhD.
The bottom line
Find a topic you are really interested in for your Master's thesis, regardless of when you will be working on it. If you know the topic beforehand you can at least read up on it and discuss preliminary aspects of the project in your PhD applications.
Admissions committees want to see research experience (which in university number 1 you will get from the semester projects) and enthusiasm/ motivation for the topic, which you can demonstrate by talking about the ongoing thesis work.