I'm sorry, but this is a horrible advice. It sacrifices communication for the sake of enforcing an arbitrary aesthetic.
Until mid-20th century, mathematics had never been communicated in this austere manner.
When you are a working mathematician, you never start with a definition. You start with a context in which your exploration begins.
It might be a question which someone else asked that interests you (and there is a story as to why). It might be that you don't even have a question, but the objects of your study are not studied enough, so you hope you stumble into one. You can easily tell why this is an interesting thing to look at.
You do some calculations, take a look at a few examples, see if you can make a stab at the chaos in front of your eyes and find a pattern, which we formally call a conjecture.
Then you see if this pattern holds in other cases, and why. This shapes the backbone of the proof.
Once you have a basic idea of a proof, you can formulate a theorem. In the formulation, you list all the conditions to which your proof applies. The pattern might be much more general, but your proof might work e.g. "only in cases when the order of the group is invertible in the base field", or some other.
After all the work is effectively done, you decide that a concept that appears repeatedly in your reasoning deserves to have a name. Something convenient to call it by, so you don't have to repeat yourself. So you make a definition.
You then decided to share your joy with this world, and write a paper.
You listen to the "beautiful advice", and throw out anything that makes your paper interesting.
Context goes out of the window, along with any hope for the reader to have any idea why your paper is worth looking at. At best, you'll advertise this in talks, or explain over beers. Side note: you will have to drink a lot of beers to make it in math.
Then, you follow the advice again, and lay things out in the order exactly opposed to the one you were thinking in: definition - theorem - proof - conjectures - examples - context.
Wait, you already scrapped context, and the examples you started with aren't illustrating the results you ended up proving.
So you tidy up your paper, come up with more specific examples, and remove anything that wasn't on the direct path to your result.
Having climbed to a place where you can see better, you pull the ladder up. Good luck to anyone outside the group of five people who are actively working in this niche!
And finally, you write an abstract to your paper, where you mention the things you defined. The abstract doesn't make any self by itself, one needs to be in-the-know to get half of it, and read your paper to understand it.
In practice, it acts as a "No Trespassing" sign for the outsiders (i.e. anyone not in direct contact with the five people you have beers with at the Annual Niche Field Conference).
Satisfied, you lean back and post it to arXiv.
It's been a beautiful day, you think. This practice is difficult to enforce, but you kept it as an aim, and got pretty close to perfection (as exemplified by a Bourbaki text, or anything by Serge Lang, but I repeat myself).
Somewhere not too far away, a student in the class you're teaching cries.
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I said "you", but as someone who's written a couple of math papers, that's really me too. We are all taught in a horrendously backwards (literally!) manner.
This perversion of the beautiful art isn't a new observation. I can't write better about it than Vladimir Arnold[1] (a titan whose name is, I hope, familiar to you).
It's worth a read to anyone who has ever studied mathematics:
[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html