Definition of basement

interesting that you have the garage entrance and front door on opposite sides. What does the site plan look like?

Regarding the basement floor: If you add some fill next to the fitness room, you won’t have problems with the 1.40 m. I estimate the garage as part of the basement, but that would have to be clarified with the authorities.
Concerning the floor height, you could work with different ceiling heights to achieve a comfortable height in the fitness room – after all, it is about an average height. Fitness 2.35 m, rest 2.15 m should be possible.

The issue with the arrangement of the front door and garage driveway has already been discussed extensively. The background is simply that the terrace to the south side can only be at ground floor level if it is elevated or mounted. If you move the double garage to the north side and elevate the terrace, you would shift the house too far south and leave the building plot. Since the front door in my opinion also belongs to the ground floor (a side entrance door not shown in the visualization is planned in the basement), I believe only this version remains. There are certainly different aspects and views, but hillside construction does not allow uncompromising building.

Regarding the issue with the room height:

The definition is a full storey is a storey above ground level which has a clear height of 2.20 m or more over at least half its floor area.

It does not say anything about an average height. Therefore, I do not think the heights may be averaged.

However, the question remains open as to exactly where the measurements are taken from and to. If from the top edge of the flooring to the bottom edge of paneling/suspended ceiling, one could theoretically simply reduce the clear room height with such measures to 2.20 m?!

Here it looks as if the garage driveway slopes downward. Then you have a kink at the point of entry into the garage, where height must be taken into account.
If the garage driveway is level … then it should still not be underestimated. Quickly googled sectional door dimension drawing:

As I said, an SUV is neither present nor planned, so I do not think it will be problematic with normal cars.

The definition is A full storey is an above-ground storey that has a clear height of 2.20 m or more over at least half of its floor area.

Well, then it's simple.

There is nothing about an average height. That is why I don’t believe you are allowed to average the heights.

No, you are not allowed to average. So not arithmetically. You can still do different heights. Constructively complex, but if it is worth it to you.

If the finished floor covering up to the bottom edge of the panelling/suspended ceiling is OK, one could theoretically just reduce the clear room height with exactly these measures to 2.20 m?!

Yes, that is possible. But there should be plausible reasons visible as to why it was constructed that way. If it is clear that the whole thing only serves to circumvent regulations and the construction can be removed again within half a day, you will probably get visits from the building inspector more often.
From which side is there an objection to the planning? The visualization gives me the impression that 1.40 m would not be exceeded. Have you calculated exactly? Same with the attic: roof pitch <45° and no knee wall – are the 2/3 exceeded? Even if so, it cannot be much that exceeds it. Show the full-story evidence. From the house, a planner should be able to easily make a single-story one, especially since you only need normal windows on the garage side.

especially because you only need normal windows on the garage side.

However, the visualization is not correct. In addition, there are supposed to be doors to the basement "up front." The question was probably whether the garage belongs to the basement, as it contributes a lot to the free wall area (1.40 on average).

The visualization is not correct.

I cannot work like this

whether the garage belongs to the basement

Yes, it is part of this floor. With some negotiation skills or a detailed explanatory report, the designer might manage to exclude the non-integrated part of the garage – e.g., with a comparative calculation of an integrated and a standalone adjacent garage.

Regarding the basement floor: If you build up the area next to the fitness room a bit, you won’t have any problems with the 1.40m.

I’m afraid that applies as measured from the original terrain.