Li-Ion batteries are happiest when their charge is between 20% and 80%, ideally at 40%. They lose a greater amount of their life when at a high charge or a low charge. It is not reasonable to only charge the battery up to only 80%, but it is very reasonable to not allow it to discharge to below 20%. If you are still carrying a brick phone then the idea of full discharges is still valid. If your phone is less than 10 years old then read the battery university article.
Ok you got me! I suppose I mis-stated when I said I fully discharge my phone! What I meant was, I use the phone until it starts to tell me to charge it, typically 15 - 20 %. I don't think it's ever been run to 0% charge. That being cleared up, I still get diminished capacity with older batteries vs new batteries.
As for the articles I posted, they were to describe the process of charging and discharging battery (as opposed to an engine burning fuel). The SA article is a bit dated, but the principles haven't changed and the Wiki deal with all types of batteries, including Lithium Ion and Lithium Ion Polymer, which is what my Droid came with.