As you have found out, the emissions standards for these CAZ are a complete dog's breakfast and our supine governments and officials couldn't care less. However, perhaps the key statement is here on the London LEZ webpage: -
The LEZ is enforced based on manufacturer declared emissions of the vehicle rather than age, but:
Euro VI (NOx and PM) engines are generally those first registered with the DVLA after 1 January 2014
Euro IV (PM) engines are generally those first registered with the DVLA as new after 1 October 2006
All vehicles which meet the ULEZ standards will also meet the LEZ standards
On addition, Birmingham state in Section 1 5th para.that if you believe your vehicle is compliant you should contact the DVLA.
If it were me, I'd not hesitate and would immediately register an appeal with the Traffic Penalty Tribunal on the matter, because this sort of thing needs sorting out. I do note, however, that Birmingham are no re-offering the discount whereas they weren't previously. Maybe they've been overwhelmed by appeals to TPT and somebody in their parking dept said, "re-offer the discount and 99% of people will pay".
Birmingham and Bristol seem to be the odd men out of all the non-London CAZs, so one has to wonder how their standards differ from everybody else.
https://vehiclecheck.drive-clean-air-zone.service.gov.uk/air_zones/compliance