I agree and would add that 'exchange' is the most formal in tone of the three, though it's also commonly used, especially in legal or business contexts. The stock market is sometimes called the stock exchange, and there are currency exchanges, and commodities exchanges such as a livestock exchange -- all venues where professional brokers make deals in large quantities. A discussion can be an exchange of ideas. The vacation deal you mention is usually called a home exchange. (It sounds like fun, by the way, but I would guess it only works if you live in a city large enough or scenic enough that many tourists want to visit, or where there are places like universities or museums that need temporary residences for visiting artists or academics.)

In fact, those examples remind me that the noun may be more common than the verb, though the verb is also used. Most commonly, a customer who is not satisfied with a purchase, or a gift, may return it to the store and exchange it for something else.

In some contexts where 'exchange' might be used in writing, we would often just say 'change' or 'trade' in normal conversations. For example, if you have a group of people for something like a photo or a choir concert, you might ask two of them to change places / trade places (with each other), so that the shorter person was closer to the front or whatever. On an overseas trip, you change money into the other currency.

'Trade' is the most common in everyday conversation, at least in AE. Children trade toys with each other, and collectors trade baseball cards, stamps, World Cup stickers, etc. If you don't like your dessert at a restaurant, your kind spouse might offer to trade with you. When you buy a new car, you often trade in your old car. ('In' there is a particle, belonging to the verb, so is accented.)

In history, merchants were traders, trading in spices, metals, slaves, jewels, whatever. ('In' there is a preposition, so is unaccented.) In North America, Canadian fur traders were among the earliest explorers, and in the Wild West, there were traveling horse traders and cattle traders. Even now we still speak of trade in the context of illegal selling, such as the drug trade or the ivory trade.

'Trade' is used in contexts such as international negotiations over tariffs, open borders, etc. Traditional capitalism usually favored free trade. The US and Canada are trying to strike a new trade deal. Progressive economists want fair trade more than free trade, with protections for workers and the environment. A proposal for a kind of free-market carbon tax in the US has been called 'cap and trade.'

'Swap' is considerably less common than the others, at least in AE, where it may sound old-fashioned or countrified. There is or was evidently in some time or place an event called a 'swap meet,' where people just bring things they no longer want for others to take. Nowadays Americans usually have garage sales or take things to Goodwill or the Salvation Army.

BE may use 'swap' more often, and in fact I believe may have once spelled it 'swop,' which suggests a more British pronunciation.

Some of your sentences do contain an idea of exchanging one thing for another, but I don't actually think we would necessarily express them with one of these verbs.

Tausche Auto gegen Fahrrad.
Ich würde gerne meine Winterjacke gegen eine leichtere tauschen.

If I were communicating that kind of idea in English, I think I would just say something simpler, e.g.,

I'm giving up my car and getting a bicycle.
In the future I'm going to ride a bicycle instead of driving a car.

I'd like to get a lighter-weight jacket instead of this heavy winter coat.
I'd like to replace my heavy winter jacket with a lighter one.

But in more formal contexts, you could still say 'exchange.'

Esau exchanged / traded his birthright for a mess of pottage.
In authoritarian countries, ordinary people exchange / trade liberty for security.

https://en.wikipedia.org/wiki/Mess_of_pottage

Maybe that will give others some points of discussion.