"Who Killed the DC Extended Universe?"

that's an old hat, and a nitpick. It's a contrivance for sure to make the concept of their team up work.

But why not say Martha? I refer to my mother with her first name when speaking about her to known strangers, such as uncle or aunt. I'm certainly not an child saying "my mom" (cringe), it's too childish.

DCEU should not be childish IMO. Also, "I don't care, he killed my mom" and "You should not have killed my mom...or squished my Walkman" are already taken by MCU. So to each franchise it's own.