Which chimney variant?

As I understand it, you cannot run the pipe through half the garden, but best along the shortest route from the chimney to the outside. If that is the case, we don't want it standing in the corner.

Well, the house probably has four sides - you can place the pipe on each side.
The surcharge for Schiedel Absolut was four digits (at approx. 12m height) - compared to the cheap version (concrete, no inserted pipe or similar), which does exactly what it is supposed to do.

Well, the house probably has four sides - you can place the pipe on any side.

Is that so? Our house has many sides because it is not square. We would have the option to place the stove in a corner where we do not want it, at about 4m distance from the chimney. Other sides where we would place it would have a distance of almost 10m. We were told that the greater the distance, the larger the required cross-section. I still do not want a pipe with a diameter of 0.5m in the corner where it can be placed aesthetically.

Is that so? Our house has many sides because it is not square. We would have the option to place the mushroom in a corner where we do not want it, at about 4m distance from the chimney. Other sides where we would place it would have a distance of almost 10m. We were told the greater the distance, the larger the cross-section required. I still don't want a pipe with a diameter of 0.5m in the corner where it can be placed aesthetically.

Therefore "probably"
For us, it is about 6-8m with a 20mm or 25mm pipe. Then it protrudes about 30cm from the ground - invisible amidst plants.
Of course, the chimney with integrated air supply is more elegant, but as said, just way too expensive.

well, way too expensive is really a big word in relation. I mean, it would have been a 2-300 euro surcharge max for me. When I calculate what the chimney and stove have already cost me as basically a nice-to-have, it doesn't break the bank anymore. But for that, it's solid and I don't have to hide anything.

Apart from the fact that a decent (somewhat presentable) intake tower is not available for under €300. The really fancy ones also often cost €600-800.

I think it was a maximum of 2-300 euros extra for me.

Then your "normal" chimney was already a very high-quality version. Maybe you could write what the chimney cost you. We were at about €1,500 for approx. 12m including inspection flaps (2 pieces) and cover plate, the inner diameter of the chimney is 25cm (square) – so rather an expensive version.