"Map cooling thermostat" is probably the thermostat heater as you are suspecting but if you didn't re-code the temp gauge the difference won't show up. The reason the temp gauge has the wide span at center is so that it doesn't move up and down when the thermostat heater (map cooling) is working properly. If the connection is corroded at the thermostat then likely it's leaking a bit and you should probably replace the stat as well.

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)