Creating a training dataset from analytical solution
I think your question depends almost entirely on the actual problem that you're trying to represent, i.e. it depends on (your) expert knowledge.
Yes, data generated this way should be considered synthetic. However there's no general requirement to add noise: adding noise (as well as how much and in which way) is normally meant to make synthetic data more realistic, i.e. closer to real data.
In general with synthetic data it's important to show that the data is representative enough of the real phenomenon it represents. Again this depends on expert knowledge but most of the time it's essential to make the distribution of the data realistic as well. So yes, if you know the true probability distribution it's certainly a good idea to make the synthetic data follow it.