Congrats. Great it is fixed. If the wheels are OE then there should more markings than just BMW. Usually it says when they were produced and who produced them. BMW does not make its own alloy wheels. Could be Borbet or one of the other OE producers.

If somebody strapped on wheels from another type of BMW with 74.1mm hubs they'd still be OEM but just not the right centre bore for your e53. Or somebody used those wheels on a vehicle that had larger hubs and had the wheels bored out to fit. Either way you are fine with the hub rings.

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2003 4.4i black ext./black int./black headliner (kid's runabout)
2014 535d X-drive, M package, silver/black interior

Sold but not forgotten:
2009 E70 35d, black ext./black int./black headliner (sold 2021)
2006 4.8iS Le Mans blue/cream int./black headliner, SOLD in 2012 sadly...

Other hardware:
2015 Cayenne S (wife's new DD and definitely wroooommmmm)
2016 Tundra 1794 edition crew cab, luxo hauler
2005 Tundra crew cab, weekend hauler. Sold after 150k very happy miles.