SOLVED
It was not easy to find the combination of alternative firmware that I tested some months ago (rik70 thanks for idea).
It was Japan ver. 2.70 (i.e. EU510_JPN270. rgn).
And I loaded it to my Nuvi 1310 by Updater tool when the device was in the pre-boot mode.
Then I started Nuvi and checked how it works and check that GarminDevice.xml is correct.
After this I loaded the original firmware nuvi 13xx ver. 5.10 ( rgn extention again) by Updater when Nuvi was in pre-boot mode.
Device started, check everything good and the GarminDevice.xml file became correct !
I could check the updates through the WebUpdater now and install the last version 5.70 and additional files. Everything works fine.

Thanks to everybody ( rik70, lolypop000, Catymag...)