How can I combine flight training hours flown under various legislations in a single logbook? I am a European citizen and last year I went to the US to start flight training. I got my student license and did my FAA private written exam. I logged about 78 flight hours with a flight school.

I also flew several times with an FAA authorized instructor but I never logged those hours because I was flying part 141 and since my flight instructor was under part 61 I was afraid I could not log those hours. I do not have a private license because I had do go back to Europe.

Now I'm back in Europe but I have the chance to fly at Christmas with an instructor in the US (not a flight school). Could I log these hours on my logbook I had started in the US?

And how can I combine my US and European logbooks?
Based on what you've written, the main issue for you is TSA approval. As a foreigner, you can't receive initial flight training in the US from any school or instructor unless you have a TSA approval in advance. And you need one approval per school or instructor: if the TSA approved you for flight training at a 141 school, you can get training from that specific school only. To use another 141 school or a part 61 instructor, you would need a second, separate approval.

But, assuming that you do have (or can get) the TSA's approval then you can log the training time as usual. The instructor must endorse (sign) your logbook as described in 14 CFR 61.51(h):


  (h) Logging training time. (1) A person may log training time when
  that person receives training from an authorized instructor in an
  aircraft, flight simulator, or flight training device.
  
  (2) The training time must be logged in a logbook and must:
  
  (i) Be endorsed in a legible manner by the authorized instructor;


And your question about multiple logbooks has already been asked.