AND FINALLY I HAVE UNDERSTOOD IN WHAT THE REASON!!!
The guys from Valve forgot to add a check "if (host_client->fakeclient) return" for functions SV_GetTrueMinMax and SV_GetTrueOrigin, which sets predicted variables 'mins'/'maxs' and 'origin' to pmove->moveents, but these variables aren't correct, as they aren't calculated for bots! - this is the source of the problem.
To fix this you need to delete these lines:
Code:
SET_CLIENT_KEYVALUE (index + 1, infobuffer, "cl_lw", "1");
SET_CLIENT_KEYVALUE (index + 1, infobuffer, "cl_lc", "1");
as functions SV_GetTrueMinMax and SV_GetTrueOrigin do checks and on these variables.
THESE VARIABLES DO NOT NEED FOR THE BOT!
P.S. also to test it on the fly, you can simply set the value of console variable "sv_unlag" to 0!