Unfortunately my problems came back. I cannot really boot anymore, not sufficient power from battery and charger only does not work.
I bought a 3-cell battery with the same specs as Purism’s from akkubuy.de, since they got a good rating on trustpilot.
Came (obviously) from China.
At first the notebook booted with the battery. But I think I made it worse with trying to get a replacement battery.
After having the battery charged one night, librem 14 is not booting with only charger, with only battery and with both attached.
It is only booting with old 4-cell battery and charger. Problem is that the 4-cell battery + charger is unreliable, because the battery does not really get charged.
I have checked this thread.
I don’t think I need the resistor for the 3-cell battery, as I got the same board version as FrankyFlawless in this thread:
It could have something to do with EC firmware, because sometimes the charger (any charger, also USB) is not recognized. You can also see in the rescue shell/terminal under /sys/class/power_supply/AC/online that it is continuously switching between 0 and 1, indicating that it is switching between charger recognized and not. The charger is new and tested with multimeter; it is steadily delivering 19V voltage.
Since the laptop is not booting from charger and 4-cell-battery is unreliable to use, I tried upgrading the EC firmware via CH341A flasher. I have the same model used in the Purism guide.
I have tried from Fedora as well as PureBoot Live ISO.
Both times I got the same output not recognising the EC chip.
According to the above Purism post it should recognise the chip as GD25Q80. However, I get the output “unknown SPI chip (RDID)” (0 kB, SPI) and WRITE operaiton not being supported.
pureos@pureos:~$ sudo flashrom -p ch341a_spi
flashrom v1.2 on Linux 5.10.0-23-amd64 (x86_64)
flashrom is free software, get the source code at https://flashrom.org
Using clock_gettime for delay loops (clk_id: 1, resolution: 1ns).
Found Generic flash chip "unknown SPI chip (RDID)" (0 kB, SPI) on ch341a_spi.
===
This flash part has status NOT WORKING for operations: PROBE READ ERASE WRITE
The test status of this chip may have been updated in the latest development
version of flashrom. If you are running the latest development version,
please email a report to flashrom@flashrom.org if any of the above operations
work correctly for you with this flash chip. Please include the flashrom log
file for all operations you tested (see the man page for details), and mention
which mainboard or programmer you tested in the subject line.
Thanks for your help!
No operations were specified
I have written to support on 1 August but have not heard back yet 
Does anyone has an idea what else to do?