This is a complicated issue! Yes, there are rules about contact after termination. It may vary from state to state and from country to country.
For example, there are quite strict rules in the state where I live how many years need to be between termination and relationship especially a romantic relationship.

The problem with contact after termination is that while in therapy there is a helper/helpee dynamic. Usually the therapeutic relationship is very one sided as it focuses on you, the client. So hardly ever a REAL friendship emerges that can be taken outside the therapy room. Even though the relationship is real and both parts have real feelings towards each other, it still is predominantly one sided and the goal is to equip us with tools and confidence to live a good life after therapy ends. Many people who want contact after termination want it because of the one-sidedness (sorry, bad word) because they still want the support, the empathy, the love and the good feelings they had in therapy. Which is why many therapists "accept" some form of contact because they want to "help" the client and because of course they care.
But many ex-clients have taken that contact too far and still do. And therapists have to weigh the individual pros and cons of this contact. Mostly it is for the client's benefit!
There are a lot of ethical questions around this and for many it's not as easy as reading a paragraph in a law book.

As to the rule to greet a client only if the client speaks first.. In my therapists disclosure form it says: "If I meet you in a public setting, like a supermarket or on the street, I will not acknowledge you or speak to you."

This is actually very important because it has solely to do with the confidentiality the therapist is obliged to uphold. Like for example a client is at the supermarket with a friend and the therapist comes up and asks how the client is, or even just says hi, it may compromise the confidentiality because the friend might ask who that person was and it could put the client into an awkward situation. It's not rejection or not care, it's simply because of confidentiality and I actually appreciate that a lot.

I hope this helps.