Crane from the neighbor on my property

Such a "handshake" is legally just as much a "contract" as a 20-page document. Only the evidential value is (at least without witnesses) logically significantly more difficult.

Exactly. And where third parties can impose a burden of proof on someone that such a contract existed, oral claims are "gladly" dismissed.

Those who own property have more worries than those who do not own property - it has always been that way until now
I agree with @bortel and @Grantlhaua

As often as you agree with me lately... oops: I'm happy

I would not consider the crane placement by the neighbor on your property as your construction activity.

The owner as a person is very rarely also the installer/user of a crane. These will always be third parties on behalf of other third parties; in any case, the result is that a crane stands on your property with all associated obligations. Saying "That’s not my crane" won’t help much.

Let me phrase my opinion differently: your neighbor will probably finish on time, the crane probably won’t bother you at all, the crane will probably be set up and secured professionally by a specialized company, no playing children will probably get injured, the neighbor can probably cover additional financing if he runs out of money, and probably everything will go smoothly anyway and you won’t need any contract, whether oral or written.

It’s only about the unlikely case that something does not work out, an accident happens, the BG-Bau carries out an inspection, or whatever. And in that case, it is better if no foreign crane stands on your property and no foreign earth is deposited there.

How likely or unlikely any complications are must be assessed by each person for themselves.

I just say... I absolutely cannot sugarcoat the threat of construction delays at the neighbor’s over the winter months.

At least you get asked...
At our place, the neighbor simply dumps 100m³ of soil on our property...

But what am I supposed to do?! I can hardly mess things up with the people I will probably have living next to me for the next 50 years...

But I would talk to him. You don't have to put up with everything.

It's only about the unlikely case that something doesn't work out, an accident happens, BG-Bau conducts an inspection, or whatever. And in that case, it's better that no foreign crane stands on your property and no foreign soil lies there.

Do you insure every most unlikely case that could happen?
In this crane case, in most possible scenarios, it's not the property owner who can be held liable, but the owner of the crane or the company (BU) that sets up/has the crane set up.

And then you also have to ask how many days it will be. Or was it stated anywhere? For us, the crane was set up for 2 days. It could be that it was also on the neighbor's land... no one cared. If something had happened: no one could see with the naked eye where one property begins or ends... everyone had an eye toward the church in the village.
What makes a new development area good and should also be lived: you feel like you are part of a large community of solidarity and shared interests.
Of course, that loosens up after moving in, but those who sat together in the same boat will remember later. Those who didn't were already fools beforehand.