MI'm confused (by YOUR posts so far, as opposed to senility ;-)
Your first post showed a Green Battery that was VERY small. It only extended ~ 1" beyond the brace over the battery, and the hold-down behind the battery appeared to be in the 3rd threaded hole from the rear.
Your LAST post shows a White Exide battery with 90 Ah rating that appears to be H7 or possibly even H8 in case size. THAT is the size and Ah capacity you NEED if you have to replace THAT battery.
Is the WHITE battery the one that is NOW in your vehicle?
IF SO, I would try slow charging THAT Battery (at the Jumpstart Terminals under the hood), and see if it will accept and hold a charge.
IF you have to replace the White Battery, the Walmart H8 battery I linked is the cheapest replacement of a "Correctly-sized" battery for your vehicle. $130 for a battery with 3-year free replacement, and additional 2 years pro-rata replacement Warranty. ANY Walmart sells them. Large Walmarts have one in stock.
As for the concern about proper function of the charging system, GET THE $7 DMM.
We have to get your battery charged/changed to Start the Engine, to MEASURE SYSTEM VOLTAGE with the DMM, with Engine Running and Alternator Charging (hopefully ;-)
SUMMARY of what I would do:
1) Get a $7 Multimeter;
2) Beg, borrow, Buy ($40 at HFT) a battery charger;
3) Try charging the current battery (particularly if it's the White Exide);
4) Slow Charge at 6-Amp charge rate for 12 hours, reducing to 2-Amp charge rate (slide switch on Charger) for an additional 12 hours.
4a) If you need to remove the battery to take it inside or to a shop to charge it, let us know and we can give Step-by-Step instructions on removal so you don't damage something. There are some hidden "gotcha's" to avoid.
5) Remove charger and wait 30 minutes (for "surface charge" to subside);
6) Measure Voltage at Jumpstart Terminals and record value;
7) If Voltage 12.2V or more, attempt to crank starter; if Starter just clicks or interior lights go OFF, turn off ignition and measure Voltage at Jumpstart Terminals AGAIN; if it has fallen to ~ 10.5V or less, you battery has failed internally.
8) IF you need to replace your Exide, get Walmart H8.
9) Check back for installation details.
YOU can do this if you just take the time to understand the basics, and the DETAILS/CONCEPTS. Any car you will ever own for decades will have a battery, so might as well learn how to diagnose and maintain the Power Supply and Charging System. You will save yourself a LOT of grief in the future.
There ARE some tricks to determining WHAT the Power Management system is set to for the Battery Type and Battery Capacity. BMW Software such as INPA or ISTA can provide that data. ANYONE could have changed that "Setting" with proper software since the vehicle left the factory, so ultimately that SETTING should be determined, and changed if necessary. If the setting is NOT correct for the current or replacement battery, the Starter will still crank the engine. The engine will run (if NO other faults) regardless of incorrect setting. BUT the charging system may NOT properly charge the battery.
George