How can I combine flight training hours flown under various legislations in a single logbook?
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.