der Sattel
saddle
A seat for a rider on the back of a horse or another riding animal.
Sie legte dem Pferd den Sattel auf und zog den Gurt fest.
She put the saddle on the horse and tightened the girth.

saddle
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saddle
A seat for a rider on the back of a horse or another riding animal.
Sie legte dem Pferd den Sattel auf und zog den Gurt fest.
She put the saddle on the horse and tightened the girth.

The verb 'satteln' (to saddle) is derived from the noun 'der Sattel' (saddle). It means to put a saddle on a horse. The expression 'umsatteln' (to change saddles) means, figuratively, to change one’s profession or field.
Bevor wir losreiten können, müssen wir die Pferde satteln.
Before we can ride off, we have to saddle the horses.
The idiom 'fest im Sattel sitzen' (to be firmly in the saddle) means to have a very secure and unchallenged professional or political position. The image is that of a rider who cannot fall off the horse.
Obwohl es viel Kritik gab, sitzt der Minister weiterhin fest im Sattel.
Although there was a lot of criticism, the minister is still firmly in the saddle.
You can tell which meaning of 'Sattel' (saddle) is intended from the context. Words like 'Pferd' (horse) or 'reiten' (to ride) point to a riding saddle, 'Fahrrad' (bicycle) to a bicycle seat, and 'Berg' (mountain) or 'Gipfel' (peak) to the geographical form.
Der Sattel meines Fahrrads ist sehr unbequem.
The saddle on my bike is very uncomfortable.
In connection with horses, typical expressions are 'den Sattel auflegen' (to put on the saddle) or simply 'im Sattel sitzen' (to sit in the saddle). The person riding is also referred to as a 'Reiter im Sattel' (rider in the saddle).
Schon als Kind saß sie sicher im Sattel.
Even as a child, she sat securely in the saddle.
seat / headquarters
'der Sitz' and 'der Sattel' can both have to do with sitting. 'Sitz' is more general: it can be a normal place to sit, for example in a car, on a bus, or on a chair. 'Sattel' is more specific and means the seat on a horse, bicycle, or motorcycle. In riding, 'Sitz' can also mean the person's posture, not the object itself. The words can be confused because you sit on a saddle. If you mean the part on a bicycle or horse, say 'Sattel'. If you mean a place to sit in general, say 'Sitz'.

Vor der Radtour stellt Sofia den Sattel etwas höher.
Before the bike ride, Sofia raises the saddle a little.
Der Sitz im Auto lässt sich nach vorne schieben.
The seat in the car can be moved forward.