I need help clarifying the term forecast interval I am reading Introduction to Time Series Analysis and Forecasting by Douglas Montgomery et al.
They describe the term forecast interval as:

The forecast interval is the frequency with which new forecasts are
  prepared. For example, in production planning, we might forecast
  demand on a monthly basis, for up to three months (the lead time or
  horizon), and prepare a new forecast each month. Thus the forecast
  interval is one month, the same as the basic period of time for which
  each forecast is made.

Generally forecast interval is used interchangeably with prediction interval. However, in this case I have hard time understanding what exactly it means. Is it the same as one-step prediction vs multiple-step prediction?
The part of the book (page 5) that discusses this does not include an example.
 A: Interesting thread and confusing terminology.  Let's say you have two researchers who are trying to forecast the number of salmon which are returning to spawn in a particular river.  
Researcher A is only interested in forecasting this number for the year 2019. Using historical data on the yearly number of salmon who returned to spawn in that river from 1990 to 2018, he will forecast that 2,000 salmon will return in 2019. To give an indication of the uncertainty associated with his point forecast, he will also provide a 95% forecast interval of 1,000 to 3,000 salmon. 
Researcher B was hired to produce forecasts and associated measures of uncertainty (i.e., forecast intervals) not just for 2019, but also for 2020, 2021 and 2022.  For the 2019 forecast, Researcher B will use the same historical data as Researcher A. For subsequent years, Researcher B will use updated historical data as follows:
2020: historical data from 1990 to 2019
2021: historical data from 1990 to 2020
2022: historical data from 1990 to 2021
So Researcher B has to wait until 2019 to produce the 2020 forecast, until 2020 to produce the 2021 forecast and until 2021 to produce the 2022 forecast.
Now, let's assume that Researcher A gets gelous of Researcher B and decides to use the historical data from 1990 to 2018 to create point forecasts and associated confidence intervals for the years 2019, 2020, 2021 and 2022. The resulting point forecasts are called one-step ahead, two-step ahead, three-step ahead point and four-step ahead forecasts, respectively.  Similarly, the associated forecast intervals are called k-step ahead forecast intervals, where k = 1, 2, 3, 4.
In contrast, Researcher B will produce only a one-step ahead forecast in each year.  
Montgomery refers to the frequency with which Researcher B will make his one-step ahead forecasts as the "forecast interval", which is totally confusing, as forecast interval is usually used for the uncertainty interval we build around the point forecast itself.  I agree with what was suggested here that forecast construction frequency constitutes better terminology.
Hope this concrete example adds a bit more clarity to the issue you raised.
A: Forecasting interval in this case would mean how often you produce the forecast. What you're describing can be called the periodicity or frequency-- where "higher" and "lower" frequency would refer to shorter and longer periods of time (eg, higher=daily or shorter, lower= monthly or longer).
A: from my version of the text (2nd edition) " One way to do this is to provide a prediction interval (PI)
to accompany the point forecast. The PI is a range of values for the future
observation, and it is likely to prove far more useful in decision-making
than a single number. We will show how to obtain PIs for most of the
forecasting methods discussed in the book."    
Forecast interval is the "bucket size" i.e. day,week etc ... whereas prediction interval reflects the uncertainty around the expected value..
Forecast intervals are often naively assumed to be symmetric and free of anomalies. Modern software using re-sampling of the error terms and the historical frequency of anomalies for the particular forecast period are definitely the way to go.
