The main theme throughout the various forms of the infinitive and the -ing form is that the infinitive suggests completion and the -ing form indicates continuousness. You can see this most obviously in the tense forms ie simple and progressive (continuous).
In your examples have someone do or have someone doing the same theme persists. The idea of possibility or near possibility is not relevant. It is the sense and the meaning of the uses of these two forms.
If you have someone do something, the sense is that they will do it or have done it.
If you have someone doing something, the sense is that the process is ongoing - the picture if you like is the activity rather than the act.
The way I’d personally tend to use/understand these two sentences might be as follows:
1- He had me wait for two hours.
For example:
He = dentist (maybe even Alan’s dentist :lol:)
I went to the dentist and had some work done on my teeth. I asked the dentist if I could eat right away after the appointment. The dentist said no, I should wait for at least two hours before eating anything. He had me wait for two hours.
2- He had me waiting for two hours.
I had an appointment with the dentist for 9:00 a.m.
I arrived at 8:50. Nine o’clock came and went. The dental assistant told me that the dentist had had an ‘unavoidable delay’ but that he would arrive shortly. Ten o’clock came and went. By 10:30 I was beginning to get a little hot under the collar. At 11:00 the dentist finally arrived.
When I later told this story to friends I simply said:
“He had me waiting for two hours.”
I think my examples pretty much reflect what Alan described.