John, every advice from you is taking me further, but I am still not quite there yet.
Now I have another error message:
'avrdude: Yikes! Invalid device signature.
Double check connections and try again, or use -F to override
this check.
Error while burning bootloader.'
I can see this error in both cases: for LilyPad and for standalone Atmega 328 and on both IDE's: 1.6 and 1.6.1.
Do you have any idea what to do next?