The techniucal term for the UK (and by extension, Commonwealth countries) is 'Constitutional Monarchy'.
As a description it is apt to cause confusion has the UK has no codified Constitution, but the term refers to a situation where the Head of State is a Monarch but the country is ruled, and hence the Monarch limited, by democratic means.
In just about every use of the word Democracy as we use it, we refer to representative Democracy, where the only power the people have is to be able to elect those who represent them, not having direct power in of themselves. This is so much the case that we note the exceptions as different, and treat representation as the standard definition of Democracy.
Although it is a word where it is very difficult to get everyone to agree on a definition, the term 'Republic' often refers to a coalition of smaller states forming a larger one, generally in a democratic manner. Its historical roots certainly come from this and it is certainly the point at which the US started using the term universally. Its hazy interaction with the words 'Commonwealth' and 'Federation' does not help, especially as that ties us back to Constitutional Monarchy.
You won't find a single objective definition of Republic though. Basically, it is what countires claiming to be one say it is, and bearing the history of the term in mind.
Its only real link with being somewhere without a monarchy is that it was the name places tended to use after they stopped being one. No other logic there at all.
The linguistic root of the word is 'public interest'. Which is basically no help at all as all Governments strive to do that, again with us noting the exceptions. Unless you are a conspiracy nut.