DIY Power Bank For Router & Fiber ONU

I’m trying to build uninterrupted power bank for my 9V 1A router and 12V 0.9A ONU. I have made a circuit Mindmap. But I think I have done some questionable things in it. Such as, I don’t know the efficiency of the LM317 based 12V to 9V linear voltage regulator. I think I’m loosing 50% of power during the conversions. So there isn’t much power to run this devices and charge the 18650 lithium battery simultaneously.

Joint box means only wire joints.

|<, >| This are diode symbols.

On 2nd mind map the relay cut’s off when the battery is changing.

Which one is better than other from the above? What can I do to improve the better one even more?

Is there any good quality diagram exist for this application?

What type of diode should I use, like, zener diode?

Also I want to build this as cheaply as possible. Thanks in advance for the support.