Who would certify the log book of such an student or licence applicant?
So, it depends. In the easy cases, it would be the Chief Flight Instructor or, in the case of an Air Operator, the Chief Pilot. TC is only about 30 years or so behind the FAA in the way they do things. Whatever the FAA or JAA did in 1990 is what TC is redoing as the latest and greatest! We're so far behind the world we think we're first! I'm joking of course. But not by much.
For the rest of this, a bit of context: In Canada, we use the term CFI to describe the Chief Flight Instructor, not just any ol' certified instructor as they do in other locales.
The CFI compares the PTR to the pilot's logbook. Upon seeing that they were within an acceptable range, they apply a stamp or written statement to the most applicable page in the logbook that reads something along the lines of "(All) Times certified correct on [Company Name] Company Aircraft from [DATE] to [DATE]" or "I hereby certify that [STUDENT] has undertaken [TRAINING] as logged." Then signed by the instructor along with their printed name, the date, and their license number. The CFI cares about accuracy because if you send too many pilots to a flight test without the proper credentials, you eventually get a visit from TC.
Now, in those cases where an instructor is not affiliated with a flight school, it will be the instructor who signs the hours in the logbook. As for which instructor if there are multiple instructors: I don't remember there being a hard and fast rule, but I used to interpret the regs to mean the instructor who is either sending the student to the flight test or who is authorizing the student to now hold the privileges if no flight test is required. However, when I was coming up the ranks, unaffiliated instructors were not as big here in Canada as they were in the US. This may have changed in the last 20 or so years, so I'll defer the correct answer to Big Pistons Forever if I'm wrong.
For commercial operators, it is the Chief Pilot who certifies the logbook. But instead of a training record, they compare the company records to the pilot's logbook; although it's been my experience here that most just look at the total time and sign away as long as it's not in gross error - they generally don't care if it's that accurate as the only time a working pilot gets their logbook certified is when they're applying for their ATPL or another job (or both!).
I'll also note that certified logbooks were also the rage when I did my PPL conversion and CPL training down in New Zealand back in the early 00s, so this is not a uniquely Canadian thing.