Single-family house with a granny flat on a 450 sqm hillside plot

Here are the ideas I have come up with based on your suggestions. I have assumed a base area of 10.75 x 8.75.

Regarding the floor area ratio, we should not have any problems at all, because according to the development plan we can build two full floors.

That is probably not quite correct. You are allowed to build two full floors. If you do, however, they may only have a total of 180 sqm. So, for example, 2 floors with an external dimension of 9x10. That is very tight for 4 children. Therefore, I wanted to make the upper floor into an attic floor so that it is no longer a full floor. Then the base area practically doubles if you want, but you then have the roof slopes in the attic.

The goal is to build only 2 floors and shed ballast to reduce costs – but still be large enough to accommodate all the children and rooms.

The terrain modeling is still completely missing here. That is very time-consuming for a quick sketch. So you have to imagine it. You don’t fall out of the terrace door at the top; instead, there is fill up to that point. ;)

I understand your comment to mean that we could plan two separate residential units on the two levels you proposed (building with a basement, foregoing the 1st floor) (at least pre-installation for a second, autonomous residential unit) so that later, when there is less space needed (children moving out), one unit could be rented out.
But I might also be missing something. The night with the twins was tough.

Exactly, you are missing something. The point was that families with twice the average number of children can expect a fruitless search when they scour house catalogues for suitable models and then understandably resort to self-designed plans; on the other hand, catalogue designs with two apartments offer two by two children's rooms, two bathrooms, etc. without major changes. If only little has to be changed, the catalogue model is just as effective a way to build cost-consciously as the "foregoing" of bay window frills. I would never recommend something like an appendix with prettified rental potential – from my point of view, that is a bug, not a feature.

Our currently available offers range from 540,000 euros to 620,000 euros for the construction of the house taking into account our requirements. Exterior facilities and earthworks would come on top of this.

Since those would indeed come on top, they have to be deducted accordingly from the 550k budget. Assuming we calculated with 480k, adjusting would mean going from the 620k offer price to a design sized 48/72 (that is one third smaller). Putting one children's room in the attic and postponing it for later – but needing a proper staircase there, etc. – does not "save" you here.
Write down the control value for how big each room is in your current apartment!
The double upgrade, first to ownership and then also second to everything bigger, does not succeed at the usual cost burden!
And forget your stupid "reserve," it is not shown in the budget.

The costs were manageable and amounted to less than 1/6 of the sum for service phase 5,

The design refinement is service phase 3 and is therefore anyway only 2/3 of the sum for service phase 5.

... then adjusting would mean going from the 620k asking price to a 48/72 large (i.e. one third smaller) design. Moving a children's room into the attic and postponing it - but needing a proper staircase there and so on - won't "save" you.

The question that torments me: is it cheaper to build 3 stories but compact, and therefore have no sloping ceilings on the upper floor, or better 2 stories with sloping ceilings on the upper floor? What would you estimate, what is the financial difference? I would feel the 3 stories would still be more expensive. But it can't just be the staircase that makes it so costly. On the other hand, with more area here you also have to move more earth. I'm really torn. If there's hardly any difference, maybe one would rather build compact but tall.

Well, without a development plan - but I already said that yesterday, is missing here - this is all hypothesis and speculation. A rafter roof - additionally with an attic/loft ceiling as a "collar beam" - is probably more expensive than a third stair landing and a simpler truss roof.

Well, without a development plan - but as I already said yesterday, is missing here - it's all hypothesis and speculation. A rafter roof - moreover with an attic/loft ceiling as a "collar beam" - is probably more expensive than a third stair floor and a simpler truss roof.

On the other hand, the additional underfloor heating + floors on another level, ceiling, stairs. Really hard to say. If it gets too compact, it looks like a tower. That's not nice either. Dumb. One could also ask whether it would be allowed to exclude individual rooms from the floor area ratio - for example, the utility room and stairs. Maybe they can be persuaded with a tearjerker and four kids under arm?