Glad u've had a pretty positive result and pretty quickly too.
Bear in mind im not a 'proper' tree surgeon ( althou spent years on the railways butchering 1 side of trees) but i thought words like 'topping' were not really best practice anymore.
A lot of councils still pollard roadside trees, which is essentially topping them, but once u start u have to continue, but generally considered bad practice to start now and not on a beech, most tree surgeons talk more about crown uplifts and crown reductions now, if done well should not create future work the way pollarding'topping does.
The other thing i just had a better look at ur photo and seems to be a big V fork about 10ft up (assuming thats a 6ft fence in background) with a big limb coming out of it, from the photo looks like a bark included union and would be a very likely place for rot to start and build up, from the photo which can be misleading doesnt look a great join and quite long, it does look like atleast its leaning away from ur property thou.
Did the tree oficer metion it or stick a ladder up to inspect?
Might be worth sticking a ladder and having a look/wee poke about with a stick see if any holes, assuming its safe to do so,
I may be wrong as not easy to tell from a photo but might be worth mentioning it for a special look as a likely bit to fail if the tree was to fail