Floor plans of a single-family house on 640m² with low eaves height

The price quoted for your requested program sounds realistic.

We will come back to the floor later.

This is a cheerful price guessing game. Rarely have architects overestimated upwards. Don't just think about where you want to live and which rooms should be included, but also how your life unfolds. What routines, preferences, dislikes, requirements for quiet, sociability, privacy (also for the children), connection to the garden or the view, light, warmth, hobbies, functions, details... you have. Gather all this and create - as your wife has already started - a gallery of images with everything you like and describe why and what you like about it. Discuss this collection with the architect and let them know that you are fully aware that not everything fits together in one house. Then set the budget (not the full amount) and let the architect get creative. Fewer square meters might be enough, or you might need more. If the architect is good, they will be happy and create a design that will surprise you - ideally in a positive way. What a good architect can come up with on a hillside site exceeds the imagination of us "one-dimensionally thinking" laypeople. That should definitely be taken advantage of.

Don’t get stuck living downstairs and sleeping upstairs. Maybe it makes sense to cook and eat downstairs and live upstairs with an open gable, because your wife likes to read and you want to watch football with your buddies using a projector and screen. Check out model homes. There won’t be any that fit your plot, but you’ll learn about dimensions and it’s a good time to check your partner’s no-gos. It’s too late at the tile exhibition. Walk through all the apartments you see with open eyes. Put together your space program for the architect. Approximately like this - Coat rack 3m shoe cabinet - Living room 4m floor-to-ceiling bookshelf - Dining table at least 1x3m bench is a no-go - Kitchen with island 1x3m - Bedroom 9m continuous wardrobe - Office is the gaming room, main thing is quiet, 8 power outlets and LAN. - Guest WC normal hand basin etc. What is important to you, where you differ from others.

So far, we have considered bringing the basement forward in order to extend the terrace like a kind of balcony in front of the house on this offset

That's exactly how we did it I can absolutely recommend it.

Eaves height valley side: 3m
Eaves height mountain side: 6m

You got something mixed up. The valley side may be higher because it’s measured from the ground

Space requirements on the ground floor: cloakroom, guest WC, open living-dining area, another room (playroom, office (family use), guest room)
Space requirements on the upper floor: 2 children's rooms with children’s bathroom, 1 master bedroom with dressing room and private bathroom
Office: family use on the ground floor, home office in the basement

I see it similarly to : don’t get hung up on upper and ground floor because of the slope.
You have a slope to the east. The street is southwest. You are only allowed one story (tell your girlfriend).
From the street, I see a one-story house with a gable roof; at the back it then goes into the basement level, which may protrude a maximum of about 1.40 meters on average due to the single-story restriction. Therefore, I would plan a second terrace to the bedrooms in the basement on the east side. Your desired upper floor becomes a basement with utility rooms.
On the ground floor then the main living area with a terrace on the short side (south).
The garage still needs to be considered.

approx. €580,000 for house + outdoor facilities (so the complete package)

What about incidental construction costs? Are they included?
Otherwise, with 2 stories (as I suggested) and a detached garage or carport, you’ll make it with half a million.

On the other hand, there was a strong need to communicate.