Not long at all. Usually 1 minute is enough.
If it starts to turn brown, it's been in there too long.
North American cookware companies seem to use the terms "saute" and "sauteuse" interchangeably, but technically, the saute pan is the straight-sided one, and the sauteuse or "fry pan" is the slope-sided one.
In French cooking equipment terms, the straight-sided one is called a "sautoir", and the sauteuse has higher sides and while angled out, they are not curved. The curved, shorter sided one is called a poele (and probably what most North Americans would identify as a "frying pan"). So the issue is really that there are a lot of different names floating around for just 3 types of pans.
When you saute, you want to maximize cooking surface area, so that things don't steam. Sloped sides will reduce the cooking surface, so that's why a saute pan will have straight sides. Regarding the lid, it would be used in the case that you want to cover the pan after a sear, to braise or steam. It also makes your pan more versatile.
Regardless of the etymology of "saute", according to La Cuisine de Reference, the definition of saute is to cook quickly in a small amount of oil, and any of the aforementioned pans can be used for this cooking technique.
"Sauter - Cuire rapidement des petites pièces de viande, de poisson ou de légumes dans un sautoir, une sauteuse ou une poêle, avec un peu de matière grasse."
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I've sauteed baby spinach many times and I've never had it become slimy. I've never seen a cookbook or article that stated that either.