Learn English – I had something done vs I got something done

You are right.

I had my hair cut = “I caused my hair to be cut”

In this causative construction cut is a past/passive participle and hair is its Patient, the entity acted upon. Compare the use of a verb whose past/passive participle is distinguishable from its infinitive:

I had my speech recorded^PaPpl = “I caused my speech to be recorded”, not
^∗ I had my speech record^Inf.

Note that if the HAVE NP VERB construction is employed with VERB in the infinitive, it does have an active sense; NP is taken to be the Agent of VERB, the entity which performs the action, and if VERB is transitive another NP acting as Patient/Direct Object is required:

^∗ I had Mario^Agent bring.
^okI had Mario^Agent bring my notes^Patient.

This may be ‘passivized’ by employing the participle:

I had my notes brought by Mario, which is the same construction as
I had my hair cut [by the stylist].

^∗ _{marks an utterance as ungrammatical}

The meaning of the two is often similar, but sometimes not. It depends a lot on context.

In a situation where somebody is in a position of power over someone else (for example, in a work setting, where a boss is telling one of his subordinates to do something), they can often be used interchangeably, because they can both be taken to mean to make someone do something.

In a situation where there is not an obvious difference in authority/power between the people involved, their meaning is often closer to the definitions you listed: To "have someone do something" generally implies that you're making a request (and that they accepted or that you expect that they will accept the request), whereas "get someone to do something" implies more of forcing or persuading somebody to do something. Even in this situation, however, sometimes people use the phrases somewhat interchangeably.

Note that both of these phrases are most commonly used either in the past tense or future tense (i.e. "I had John clean the windows" or "I will have John clean the windows"), instead of present tense ("I have John clean the windows"), unless you are talking about a recurring thing ("I have John clean the windows every day"), because both requesting and making somebody do something are basically instantaneous actions in English, so your examples 1 and 2 both sound a little odd just for that reason. In my answers below I've changed the tense just to make it sound more natural:

In the examples you presented:

1a. I will have John clean the windows.
1b. I will get John to clean the windows.

If you're in a position of authority over John (you're John's boss), these both mean pretty much the same thing. If you're not (for example, John is just a housemate with equal standing to you, but you still want him to clean the windows), then 1a sounds like you're going to ask John to clean the windows (which he'll probably do because he's a nice guy or he owes you a favor, etc), whereas 1b suggests that you're going to find a way to make him clean the windows whether he wants to or not.

2a. I will have a painter paint my house.
2b. I will get a painter to paint my house.

2b is a little ambiguous because there's two ways you can interpret that sentence. You can either be using "get" in the sense you're asking about here (make someone do something), or you could be saying that you will "get a painter" (find/obtain a painter) for the purpose of "to paint my house", that is you are going to find a painter who can do that for you.

Assuming that 2b is interpreted in the "make someone do something" sense, then these mean basically the same thing because there's an obvious employment relationship here where you are able to dictate what the painter does (because presumably you're paying him/her to do it).

3a. The boss had his secretary make copies of the report.
3b. The boss got his secretary to make copies of the report.

These are pretty interchangeable (employer/employee relationship).

4a. I had the kids eat the broccoli.
4b. I got the kids to eat the broccoli.

Again, this is pretty similar. In this case, even though there is an authority relationship here, because eating broccoli is something most kids are not expected to want to do, 4b does still have a bit more of the sense of "somehow made them do it even though they didn't want to".

A few additional points worth being aware of, though:

• "get someone to do something" is a relatively casual/informal phrase, and is not usually used in formal contexts (where one usually uses something more specific like "asked", "made", etc). "have someone do something" is fine in both casual and formal contexts, though.
• In some cases, particularly in business settings or other situations where somebody has authority over multiple people, "have someone do something" can also imply that someone is choosing that person out of a group of people who could potentially do the task. "get someone to do something" can also be used in that case, but does not imply the selection aspect quite as strongly.
• Similar to the last point, in many cases, "get someone to do something" puts a bit more emphasis on the action (making them do it), whereas "have someone do something" puts a bit more emphasis on the person (that person instead of others).

So, as a result of these points, in example 4, 4a reads a bit like "I chose the kids to be the ones to eat the broccoli.", whereas 4b reads more like "The kids needed to eat the broccoli, so I made sure they did it."