Some fonts have a "Semi Bold" font style.
What is it used for?
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Edit: Based on OP comment
"It's doing what you mentioned, right? Maybe not at sizes 9, 10 and 14 (same thickness as bold), but at sizes 11 and 12 it works."
Are you asking if the file you provided is actually containing a legitimate arial semi bold?
I think you are using a file type that is too old.
FON is a font file format used by Microsoft Windows. FON files are part of the Windows 3.x font library, and contain information for fonts. FON files are older font files, which cannot be resized like .TTF fonts, and also may appear differently when printed."
I'm going to leave my original answer here which was answered literally based off your post:
What is a “Semi Bold” font style used for?
My answer to that is:
When bold is too much.
There is a whole spectrum of weights for typefaces for just the right situations.
Here they are in order.
Hairline, Thin, Ultra Light, Extra Light, Light, Book (sometimes heavier than Regular), Regular/Roman, Medium, Semibold/Demibold, Bold, Extra Bold, Ultra Bold, Heavy, Black, Ultra Black,
Maybe take a look at: https://www.quora.com/What-is-the-difference-between-Medium-Demi-and-Semibold-fonts
Because we are supposed to use as few different fonts as possible I would like even more variations in a font. The main tricks available in typography are size, weight and position. They are all used to drive attention, or more specifically to make the user read things in the order you want.
Lets talk about boldness, AKA Weight. Bolder thicker fonts stand out, look heavy and so attract the eye. If you have a big bold word down at the bottom of the page and everything else is regular paragraph text than the user will probably look down at the bottom and read the bold word first.
So bold attracts attention. Why have more than 1 bold? Because you still want to attract attention but the main bold is already used and its higher in the hierarchy of importance. You want Bold Bold read first, Regular Bold read second, because of Subtitling. You have titles, you have captions and you have subtitles. Subtitles still need to stand out.
So more weight options, more italic options, more thin options will allow you to emphasize text in many ways. This is IMPORTANT, this is Interesting, thin text is non obtrusive and falls to the back. Wide text is strong attracts attention but takes too much room.
When you have a page with several title sizes, captions, paragraphs, asides, quotes and comments, and then you need to place even another style of text, you will want a semi-bold.
Look how many variations there are for Helvetica New Lt Std (see the scroll bar?):