Yes, technically it is correct but it can't possibly be intended to work that way. What's the point of even having a search history tab when the search history is full of single letters leading up to the intended search word. Do a search for something with a long title and you'd have page after page of search history that is all meaningless except for the very last entry. It needs to be reprogrammed not to start the search until the entire search term is typed in and the user clicks on "SEARCH".
If I had to guess, I would say Dish is trying to improve the search so that users don't have to wait for the results. They can quickly see the returned shows and most if the time you will see the one you are searching for before you type in too many characters. But I totally agree that keeping EVERY character is nonsense. But without the actual "search" button to click to store what you've entered it would be tricky to determine exactly what the final term would have been. Hopefully they will come up with a solution.
My suggestion is that this is a tradeoff. Meaning, there is a save button somewhere to click AND the search process stays the same as it is now (where you start with all then eliminate choices as you type). But the only thing stored in the history is obviously what the user actually has entered so far in the search field and what they select to save.
Here are some examples of they way Dish could make search better: