HI igor brusentsev,
1. About your board which already updated the K20 bootloader firmware, and can see the "MAINTENIENCE" device.
Do you use other firmware copy to MAINTENANCE? eg, CMSIS DAP:
https://www.nxp.com/assets/downloads/data/en/snippets-boot-code-headers-monitors/k20dx_frdmk66f_if_c...
Whether it works or not?
About the JLINK firmware, you can use this one:
https://www.segger.com/downloads/jlink/OpenSDA_FRDM-K66F
copy to the MAINTENACE
After you copy it to the MAINTENACE, then you need to power off the board an power on it again.
2. About your another board
Seems your log is not complete, do you have the first log, whether the it contains the JTAG voltage?
Something link this:

Please make sure you connect to the correct interface.
When you change the SWD interface, you need to power off the board, connect the SWD debugger, then power on it again, don't do the hot connection.
If all the item is correct, you can try this JLINK commander: unlock kinetis
Whether you can do the flash mass erase or not? then try to connect the K20 chip again, please note, it need to be powered.
Wish it helps you!
If you still have issues about it, please kindly let me know.
Best Regards,
Kerry
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