Discussion on answer by David C. Ullrich: How to distinguish two groups $(\mathbb{Z},+)$ , $(\mathbb{Z} \times \mathbb{Z},+)$ using first order logic?

Imported from a comment discussion on https://math.stackexchange.com/questions/2483407/how-to-distinguish-two-groups-mathbbz-mathbbz-times-mathbbz/2485070#2485070