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.