The issue is that you're letting the battery drop below 40% - you might as well cut it in half. If you want a lithium battery to last, never let it get below 40% after conditioning it. (50% is the sweet spot for maximum lifespan.)
The phone is set to shut the phone off at 10% charge. If you then turn it on, it's at 10% (or lower) charge and shuts off. That's how it's supposed to work. (That's a "last call before you destroy the battery" notice, it's not something to do all the time.) (There are ways to set the shutoff lower, but that just destroys the battery faster - and in a phone with a battery you can't replace by just popping the back cover off, that's not something you want to do. You want the battery to last at least until you decide to get rid of the phone. (Dropping it to 10% every day will kill it in less than a year.)