In his seminal paper ``Computing Machinery and Intelligence'', Alan Turing introduced the ``imitation game'' as part of exploring the concept of machine intelligence. The Turing Test has since been the subject of much analysis, debate, refinement and extension. Here we sidestep the question of whether a particular machine can be labeled intelligent, or can be said to match human capabilities in a given context. Instead, we first draw attention to the seemingly simpler question a person may ask themselves in an everyday interaction: ``Am I interacting with a human or with a machine?''. We then shift the focus from seeking a method for eliciting the answer, and, rather, reflect upon the importance and significance of this Human-or-Machine question and the use one may make of a reliable answer thereto. Whereas Turing's original test is widely considered to be more of a thought experiment, the Human-or-Machine matter as discussed here has obvious practical relevance. While it is still unclear if and when machines will be able to mimic human behavior with high fidelity in everyday contexts, we argue that near-term exploration of the issues raised here can contribute to refinement of methods for developing computerized systems, and may also lead to new insights into fundamental characteristics of human behavior.