Yeah, the picture in the description on BMW's website is incorrect (not the first time I've seen this). My 2023 Z4 also has the buttons on the steering wheel as shown in the picture of the M240i. But yes, Active Cruise Control is the same as Adaptive Cruise Control in other makes.
Random 230i I found on AutoTrader that has it. You can clearly see the buttons on the steering wheel to adjust the distance:
https://www.autotrader.com/cars-for-...ckType=listing