1. Make sure you get the polarity correct --- never trust the colours.
2. A "6V" lead acid battery is closer to 7V fully charged.
Check how that third connection is wired on the original battery; it could be a lower-voltage tap to supply some other circuitry, or a "presence" signal, or a thermistor, or something else entirely like a custom DRM chip/1-wire EEPROM (although I'm not sure if that was so common 20 years ago.)