Battery is out of the question for my system.
Had tried many new batteries, didn't work. Had run several good hardware diagnostic software, battery status is OK. It's the RTC "clock ticking" that failed the diagnostic tests, and "CMOS update cycle not coming" problem as well.
In short, it's a mainboard problem, possibly chipset issue because both the RTC and CMOS RAM are integrated into it.
We cannot blame the battery because the CMOS and RTC get power from the system power supply when the system is powered on (not from the battery anymore). And all these problems are reported by the diagnostic tests with power coming from the power supply, not from the battery.
My power supply is also good. Have tested all output voltages with a digital voltmeter and the readings conform to the standard voltages for ATX power supply.