"Iterate" vs. "Reiterate"

Definition of iterate:

to say or do again or again and again

Definition of reiterate:

to state or do over again or repeatedly sometimes with wearying effect

The distinction seems to be that reiterate is conventionally used when the repeating of this thing is supplementary to the attempt to convey the point at hand. I don't think I've ever heard iterate used similarly. Would it be wrong or awkward to? If not, then why are there two different words? When should one be used over the other?

To me, to iterate is to do the same thing many times in mathematics or some other process. It may involve either gradually approaching a solution or going though a list.

Meanwhile to reiterate would be to say something one more time to remind a listener or reader of an important point.

No doubt they had similar meanings of repetition at some point in time, but I feel they now have different meanings and contexts for use.

iterate has to do with repeating an action - reiterate has to do with repeating verbally