a) "Box Breathing", as mentioned in the article. Inhale, hold, exhale, hold. Same rate for each step. Ex: in for 4, hold for 4, out for 4, hold for 4. Repeat as long as you like. Very calming and "centering" (in the sense of moving you toward "neutral" physiologically). More effective if you can breathe through nose and use diaphragm/belly breathing, but that's optional.
b) "Coherent Breathing". Inhale, exhale, at the same rate. Ex: a good rate is 5 seconds in, 5 seconds out. Also calming/centering, with an observed positive effect on heart rate variability. This is box breathing without the holds, so experiment to see which is preferable. Again, using the diaphragm is optional.
c) "Gratitude Breathing", for lack of a better phrase. Inhale (explicitly diaphragm/belly breathing this time); think of something you're grateful for (thankful, happy, something positive); exhale slowly. The exhale should be twice as slow as the inhale. Ex: so inhale for a count of 4, think a happy thought, exhale for a count of 8. Repeat at least ten times (to overwhelm whatever was in working memory before you began), more if you like. This will calm, relax, and trigger a positive emotional/psychological state.
d) "Relax and pass out breathing". As c) without the "positive thought". So a deep inhale, then a twice slower exhale. This is more purely relaxing than the others, can lead to sleep eventually. This does not have the double inhale a la the "cyclic sighing" mentioned in the article.
e) "Panic and freak out breathing": short, rapid inhale and exhale, using the upper chest rather than diaphragm. Why do this? Anyway.
It's been bizarre to me how many people cannot easily breathe using their diaphragms, whereas I have to consciously force myself to do otherwise. Anyway.
There is all sorts of research on these, some good and most junk, but the main takeaway is that for some reason humans have the ability to consciously control our breathing, and the way we breathe has direct impact on our resulting physiological state.
There are endless varieties of "breathwork" (Wim Hoff's stuff, actual Tummo breathing, 3721 variations of pranayama) because humans like to complicate things, but these cover the bases for routine self-care.
2) Change what you're doing. I don't know what you're doing, but the most common trigger for this in my narrow piece of the world is people pushing too far too soon. ex: shorten the count, or lengthen the count. Experiment, basically.
3) As an option to consider: ignore it and keep going. Some forms of breathing exercises make me feel on the verge of panic momentarily. It appears to be related to changes in the level of carbon dioxide where the brain mistakenly believes it is suffocating. The only two things that have worked for me are ignoring it or changing my timing slightly and building up a tolerance. (Well, a third: ensuring I am adequately hydrated.)
I only ask because there are several other commenters on here offering breathing tips, and you sound like an expert on this.