Why are(n't) rechargeable batteries damaged by partial charging?
The three statements you quote are all about correct. What's good for a battery heavily depends on the type (chemistry) of the battery.

In the old days, earlier than 10-15 years ago, the majority of rechargeable batteries were the NiMh (or NiCd before that) type. This type is more or less susceptible to the so-called "memory effect", which basically says that the battery ages sooner (starts losing capacity) if it is recharged before being more or less completely discharged.

With current Lithium-based batteries (LiPo, LiIon, LiFePo,...) which have largely replaced NiMh (replaced them completely in most areas, esp. in mobile computing), the situation is different:

Lithium batteries take some damage from deep discharges (0% or less, 'under discharge') and also suffer a bit from high charge levels (100%) especially if held there for extended periods and/or at elevated temperatures. That's why for almost all current rechargeable batteries (Lithium) the optimum useful lifespan in terms of total energy stored/provided before becoming more or less unusable is achieved by maintaining neither too low nor too high charge states. A rule of thumb is to try and keep them between 30-70% as much as you can. The actual optimum varies by battery type, manufacturer etc.; some may yield more energy in total if cycled between 20 and 80% for example, but the difference should not be much, i.e. a given battery always cycled between 20% and 80% may live twice as long as the same type of battery cycled between 2% and 98%, but going to 30-70% instead of 20-80% will probably not make much of a difference in either direction.

It may be noteworthy that the relation "lower cycles" = "longer lifespan" is not always true due to non-linear effects. That is, permanently cycling between 49% and 51% will give many more charge-discharge cycles than, say, 20%-80%, but the total usable energy will likely be less. (Notice that a "cycle" comprises only 2% of the total capacity in the former vs. 60% in the latter case. If you get 10x as many cycles for the shallow cycle, you still only get 10*2%/60%=1/3 of the total usable energy over the life of the battery.)

There are some articles at battery university, including at least one where they tested different strategies and plotted a graph of cycling range vs. total energy stored. I'll try and come up with a link.

Edit:

This should cover all your questions about what's best for Lithium-based batteries: https://batteryuniversity.com/learn/article/how_to_prolong_lithium_based_batteries

Some quotes:


  If at all possible, avoid full discharges and charge the battery more
  often between uses. Partial discharge on Li-ion is fine. There is no
  memory and the battery does not need periodic full discharge cycles to
  prolong life.


.


  Most Li-ions charge to 4.20V/cell, and every reduction in peak charge
  voltage of 0.10V/cell is said to double the cycle life.


.


  On the negative side, a lower peak charge voltage reduces the capacity
  the battery stores. As a simple guideline, every 70mV reduction in
  charge voltage lowers the overall capacity by 10 percent. Applying the
  peak charge voltage on a subsequent charge will restore the full
  capacity.


The last quote can be interpreted as: Charging only up to ~85% (which should be at about 4.1V) will double the cycle life compared to charging to 100%.

So the key points to be aware of are:


Practically all rechargeable batteries you'll find around you are Lithium-based nowadays and these should not be treated like the old Nickel-based batteries
Rechargeable batteries gradually and permanently lose more and more of their initial capacity over time
Deep discharges (like to below 5-10%) permanently reduce a Lithium battery's capacity
High charges (like to above 90-95%) also take their toll on their lifetime
Every charge "cycle" eats a small amount of capacity, even if it's only a 1% charge cycle