Why's a call option called 'call', and put option called 'put'?

An option in this context is a right to do something, so it makes sense to describe the option in terms of that right: a Put option is the right to put (sell) a security and a Call option is the right to call (buy) a security.

There are two parties involved:

• the option holder has the right to exercise the option. They could have acquired the option by buying the option, or it might have been granted to them as part of their remuneration, etc.

• the counterparty to the option holder (I'll just call them the counterparty here) has an obligation to be the counterparty to the underlying transaction if the option holder exercises the option: the counterparty must buy whatever was Put and sell whatever was Called.

Since options have two parties, you can describe them from either perspective or even both. Let's consider Put options, which you propose to define (Q2, Q4) as the counterparty "calling" the option holder to buy the security from the option holder.

Although you can describe a Put as the counterparty's obligation to buy, the transaction isn't done at their option. The counterparty has no choice in the matter. If the option holder chooses not to sell, the counterparty can't "call" the security; and if the option holder chooses to sell, the counterparty can't refuse.

You can describe a Put as the option holder’s right to sell and the counterparty’s obligation to buy. However, the connotation of the word "option" is one-sided. The derivative belongs to the option holder; the transaction triggers at their option. As such, it makes sense to describe a Put option from the option holder's perspective: they hold an option to sell the underlying security.

Likewise, it makes sense to describe a Call option as the option holder's option to buy the underlying security.

For the etymology question in the title

English often adds a preposition to a verb to alter the meaning of a verb. Calls and puts should be understood from the perspective of the option writer, along with the corresponding prepositions "away" and "to."

I have written calls. My broker will report, "Your stock has been called away." This is a shorthand for "Your (underlying) stock (on which the call was written) has been called away (and you have sold at the option's strike price to the option buyer)." (He could also say, "The call has been exercised" and hope that the option writer is well versed in the mechanics of options.) Here, the preposition away tells me the direction of the stock's motion: It is away from me, the call writer.

I have written puts. My broker will report, "The stock has been put to you." This is shorthand for "The put buyer has exercised his right to sell the underlying stock to you, the put writer, at the option's strike price." Here, the preposition to tells me the direction of the stock's motion: It is to me, the put writer.

Rather than cross-post my answer, here is a link to my answer on Quant.SE: https://quant.stackexchange.com/a/33374/343 .

For questions 1, 2, 3, and 4

The answers are no to all. You would be missing the underlying direction. For Q4, for example,

Put Option buyer is "calling" the Put Option seller to buy the security from the buyer

While it might be understandable to the OP, this unconventional usage is using the inverse perspective (of the option buyer rather than the option writer). And note that the options exchange matches up option writers with option buyers; no one calls anyone on the phone.

The put buyer "puts the stock to" the put writer. Or colloquially, (because it forces the writer to buy a stock at the higher strike price rather than the current price), the put buyer "sticks it to" the put writer.