This will depend on what you want to use the forecast and the interval for.
For instance, I do forecasting for retail, and our prediction intervals (more precisely: high quantile forecasts) are used for replenishment. In such a use case, the relevant thing is, say, a 95% quantile forecast, because this will translate more-or-less into a specific service level, which will hopefully balance the costs of understock and overstock.
As an example, let's look at the
AirPassengers dataset. Maybe we want to provide enough capacity to achieve a service level of 95%. We can do this by fitting a model and looking at a 90% prediction interval. This is comprised of the 5% and the 95% quantile forecasts. Below, the "Lo 90%" is the number such that we expect a 5% chance of observing fewer passengers, while the "Hi 90%" is the number such that we expect a 95% chance of observing fewer passengers - together, the two numbers bracket off a prediction interval that we expect a 90% chance of covering the next observation.
> (foo <- forecast(auto.arima(AirPassengers),h=1,level=0.90))
Point Forecast Lo 90 Hi 90
Jan 1961 446.7582 427.413 466.1034
So, to achieve 95% service level, we would plan for providing capacity for 466.1 (thousand) passengers. (I'm glossing over different possible definitions of the service level, which here don't really make a difference. Plus, in planning for a longer lead time, we would need to take remaining autocorrelation into account, and so forth.)
Most forecasters I work with are only interested in prediction intervals, because you can verify them to a certain extent, simply by observing the next realization. Confidence intervals can never be verified, and you will have to trust that you have a correctly specified model for the mathematics to work. (And I have never seen the tolerance interval used in forecasting.)
When you are predicting the winner of an election, a prediction interval does not really make sense, because this random variable is not numeric. Unless you are not predicting the winner, but, say, a candidate's share of the vote.