Heya - I'll try and give an answer. The tl;dr: version is "you're not wrong"

The term monarchy is indeed used to describe vastly different things today, ranging from a "constitutional monarchy", in which the monarch plays little to no role in day-to-day politics, and beyond that there are a whole range of current and past heads of states that can be described as monarchs with varying degrees of power, also I am not aware of any rule stating that a monarchy always also has to be a hereditary monarchy, even though it is mostly used in that way.

Down to the nitty-gritty: the term, like so many others, is derived from Greek "μοναρχία", which can quite literally be translate as "the rule of one", and the earliest witnesses I can find without making it a full blown research is with Athenian-centered authors of the 4th century BCE. I'll include a link to an open-acces english translation of the place in Herodots Histories book 3*, in which the author has the Persian great king Darius discuss the merits of oligarchy, democracy and monarchy respectively. If you check out the place in question, you might note that the hereditary aspect is not brought up at all. Still, one should note that monarchy is not at all the term used when describing monarchial rulers of Greek city states, here the term "tyrant" is more often applied, although at this point it lacks the negative connotation. Still one might chuckle at the thought that a polis-citizen of classical Greek might be really confused if asked to explain the difference between a monarchy and a tyranny, as these seem to be synonymous.

To the Roman Empire, though: usually the terms "Roman Republic" and "Roman Empire" are differanciated to mean the republican phase of the Roman Empire and that which follows after Augustus had brought the republican phase to an end after the civil war in 27 BCE. Many people though use the term “Roman Empire” to mean the whole shebang and really who could blame them? The republic was just as much as an empire as the post-Augustus empire, albeit smaller. Rome was certainly a monarchy in the time before the republic and before they thew out their King, commonly dated to 509 BCE, but the historiography of that period is mired in mythology, so don't expect what actually happened to be as clear cut as, for instance, in Titus Livius “ab urbe condita” written close to the end of the age. Note also that these kings (“rex”) didn’t inherit their title, they were elected into office for life. After this comes the republican time and during this period it is definitely not a monarchy, but it can certainly be argued that it became a monarchy again when Augustus rose to power, and I even recall one paper (which is itself an shortened version of a big fat publication by the same author) in which he rather unabashedly calls it a monarchy: “When Fergus Millar deliberately broke from that perspective in the Emperor in the Roman World, and adopted instead the perspective of Greek writers like Dio, his claim that the imperial system (normally referred to, for equally ideological reasons, as ‘the Principate’) was in fact a form of monarchy was regarded as controversial. I believe we can now take it for granted.“ (Wallace-Hadrill 2011,93)**

Monarchies, or tyrannies as they were more often called, became rather unpopular during the respective Greek and Roman classical periods, much more so in Rome however: for instance, the Flavian dynasty (Vespasian, Titus, Domitian, 69[nice]-96CE) was just that, a dynasty where the monarch passed his power to the next person in the hereditary pipeline, but public appearances were everything: the senate was upheld and was certainly also involved in the decision making to various degrees. Both Rome and Athens had a proud tradition of ousting monarchs – I was going to go into that but to my surprise I find that I have already spent most of the evening on this reply (and moaping around on the internet in general), so I’m going to leave it at that. This aspect of Roman and Athenian culture is also well known, but if you are interested in the monarch-ousting traditions, check out Harmodious and Aristogeiton in Athens and Brutus (no, not the stabby one) for Rome. – hope this helps!

*Herodotus Histories 3.82:http://www.perseus.tufts.edu/hopper/text?doc=Hdt.+3.82&redirect=true

**Wallace-Hadrill, Andrew: The Roman Imperial Court Society: Seen and Unseen in the Performance of Power, in: Duindam, Jeroen F. J. / Artan, Tülay / Kunt, Metin (Hg.): Royal courts in dynastic states and empires. A global perspective (Rulers & Elites 1), Leiden / Boston 2011, 91–102