The neighbor's crane site should be on our property

I think the opinion is clearly half/half ;-)

I see it that way too, as I said.

We will start a few weeks later anyway, hopefully it will work out that the crane is gone by then.

Then everything is perfect.

But with such a huge project, you shouldn't just assume that everything will go as planned: If their construction is delayed, we have extra costs:
- later move-in means paying rent longer, possibly paying interest on the loan longer, with very bad luck even a price increase from the supplier (our price guarantee expires shortly after the planned start of construction)

Appropriate contractual clauses belong there.

- if a third-party company prepares the crane site and then the crane of our construction company slips/falls/etc. (I realize that this is an absolute worst case), who then assumes liability?

Your contractor is responsible if he sets up the crane. It is his duty to ensure that the site is safe.

Actually, something like that belongs in a proper contract.

A lease agreement is indispensable here.

But that just doesn’t fit the basic attitude of most here: "have a beer, it’ll be fine"

Forget about contracts.