Firstly, I'd suggest going and speaking to the second school - there is clearly a difference in perception there, that needs clearing up. As for the reasons - there could be many, and I'd hesitate to guess.
I've flown with a lot of instructors who don't seem to feel it necessary to sign for flights in a logbook, ultimately for issue of your licence, the important thing is that they've signed on the application form to CAA saying that you've done enough hours to get your licence. A discrepancy between their records and yours *might* be relevant if one shows you've got enough hours and the other doesn't, but otherwise it shouldn't matter too much.
Changing logbooks - it's perfectly legit to buy a new and posher logbook (there have been many threads on Pprune on preferences) and copy up. BUT, because it's not your original record, keep that safe in case anybody ever asks for reference to it. I did this myself after I left the UAS and started civvie flying, and later when I started doing some military flying again for a few years I kept a duplicate record in my "posh" civvie logbook - CAA did need to see my original signed UAS logbook for my first PPL(A) application, but that's the only time anybody's ever asked to see it.
G