There is a lot of nuance in the differences, as Tᴚoɯɐuo pointed out in his comment to your question. More context would really help here. By and large, these phrases can be used interchangeably, but there are also cases where the meanings are different, and in line with the dictionary definitions that you've provided. For a simple case:
- She had him clean his room.
- She made him clean his room.
- She got him to clean his room.
These all mean roughly the same thing. In all three cases, the subject of the sentence (she) successfully directed the object (him) to clean his room. Without more context, we don't really know how they differ. We know that she provided the influence, and he obeyed and completed the task. How coercive did she have to be? We don't know. We don't even know why he complied. We only know that he did. You could infer some added meaning based on the dictionary definitions, but you could be reading something into it.
And so you begin to see that the nuance comes out largely in the context of the relationship of the two people - the one making the plea, and the one acting in response to it.
- She called the butler to her side and instructed him, "Have the driver ready the car for our trip."
In this instance, there is an evident chain of authority. When the speaker issues her command, she expects that authority to be sufficient motivation for the driver to carry it out. It is expected of him.
- (a) The young girl cried as her older sister tore off with her doll. "Make her give it back!" she squealed to her father.
Here, the appeal is made to a higher authority because the one making the plea does not have sufficient authority herself to cause the thing to happen. This would be the only appropriate choice of the three options. It can be used similarly where the person making the appeal has the authority, but does not have the ability (presently) to cause it to happen.
- (b) The young girl cried as her older sister tore off with her doll. The girls' mother, busy with the new baby, had grown impatient with their bickering. "Make her give it back!" she called to their father.
The last option probably has the most flexibility with it, and is most dependent on context, because it can be used in a slightly informal manner. For instance:
- (a) Bill knew the problem couldn't wait until he returned home from his business trip. "Get a plumber to fix the leak," he told his wife.
There's no coercion going on here. It's a natural use of the construct, and it will be a normal business transaction taking place between the woman and the plumber. Whereas:
- (b) "I don't care what the policy is here. Get him to open this door, right now!"
clearly indicates that something foul is taking place. But in this case, with the additional context that's provided, all three options would work.