I think I am correct in saying this, but whether you are modular of integrated, throughout your professional training, it is a requirement for your instructor to keep a record of the exercises you have completed and provide comments on your progress throughout the course in a written file.
With the above in mind, if you are concerned about having this information to hand, you could always request a copy of this documentation for yourself from your flying school. One thing I did, was to keep a copy of my ATPL results and CAA test assessment forms when you pass your CPL and your IR. Your examiner will give you a copy of these.
When I joined my current airline, I think the only thing that was requested was a copy of my licence. On the assessment the airline did not seem to be too bothered whether you were a modular of integrated student, they were more interested in you as a person and whether you as an individual would fit into the airline, therefore I would not get too bogged down with this integrated versus modular debate. Most airlines these days have a realistic view on this, and of course it is airline specific, it all depends who is doing the recruiting.
Its interesting now being on the other side of the fence, I am sure most people on PPRUNE will agree that we all had similar anxiety when training about the questions you are asking, its natural, however, try not to let it cloud your judgement. The good thing is, you are doing your research and asking all the right questions, therefore I am sure you will find your way.
All the best with your training, enjoy it.