Earlier quoted context omitted.
Even with diaphragmatic breathing, your chest moves eventually as your lungs fill with air. I don't think these specific instructions promote controlling your breath by intentionally inhaling into your chest; they seem pretty passive.
Yes, the chest will move very slightly in deep belly breathing. But no, this is not a trivial thing, the focus in breathing should be on the abdomen/belly, which is your center; not the chest, which is a source of tension, and especially full of tension if you're breathing up "into" your chest. Again, this is not a trivial thing, and you can easily check good sources on breathing (e.g., for meditation, singing, runni…
I have learned to control anxiety, and even fear, by simply pushing it from the chest down into the belly where it gets consumed. This was the phrasing someone used that I picked it up from, and it feels just like that. What I believe is really happening is really a realignment of breathing, just like you describe. It works.