A Tesla-coil seems to be transformer only differing in it's geometrical design?

The basic concept is similar but they are used for different purposes. Internally they are different and the output is very different. Think of a Tesla coil as the coil or coil pack in your car. Very high voltage and low current. So, starting with 12 V, you get 20,000V. With a transformer, the wiring design influences the output, typically higher amperage and lower voltages. Your house is a good example, 480 V at the pole, 240 at your house, 200-400 Amps. If you Google "Tesla coil vs a transformer", you can find the design details and how they differ. Tesla was quite a unique inventor who espoused AC voltage when Edison was for DC. Glad AC won out.