Electronic – Batteries connection to load and charging port for DIY bluetooth speaker

Look closely at the top plot and you can see it is just applying a simple maximum current OR voltage scheme. Up to about .9 hour the maximum current (looks like about 700 mA) causes lower than the maximum voltage, so the battery is charged with that current. When the battery reaches the 4.2 V limit, the charger holds the voltage there and the current then decreases. Eventually the battery gets so full that the charging current tapers off to nearly 0.

The charger puts out either 700 mA or 4.2 V, whichever results in lower voltage (or current). The battery then decides the other of the two.

You cannot use 9V/12V and plug chargers in series. You need to use 5V and connect them in parallel.
Imagine that you connect them serially to 10V, and one of batteries is fully charged and the second one fully discharged. Then the first charger won't take any current and the second one will need 1A, it is not going to work, because the first charger will block current and almost whole 10V will be on this charger.

So you have to use that 5V regulator.

UPDATE:
Your second link is for 2 cells, so yes, that is better solution than 2 separate chargers.