No airline has such a policy. A cabin bag is just that - a bag which the passenger must carry themselves onto the cabin, and store either in the overhead locker or under the seat in front of them.
If you decide to hand over all bags at check in, it means you are relinquishing your cabin baggage allowance. Anything you check in must therefore meet the limits of your checked luggage allowance, or else you must pay extra.
Every airline gives you two separate baggage limits - a carry-on allowance, and a checked baggage allowance. They are separate because you cannot mix and match the two. You can no more expect to carry on all your checked luggage allowance in addition to your hand luggage allowance, than you can expect to add the two together and check bags beyond your stated checked luggage allowance.
That this is not explicitly mentioned as policy on their website is somewhat obsolete. It is assumed that everyone will understand that anything presented for checking into the hold is checked luggage, and anything carried on to the plane is hand luggage. It's already self-evident and clear.
That Lufthansa may have done this for you in the past does not mean they will do it again, and
Lufthansa's published limits are no different from BA's for this route (2*23kg checked luggage, but
a much lower cabin baggage limit, of just 8kg (BA allows 23kg as cabin baggage).