When reading lots of articles about carbon monoxide poisoning cases I came across to it being referred as "silent killer" many times. Now, I know that carbon monoxide is a byproduct of an incomplete combustion with insufficient oxygen available in the air. Also that we as people only have a carbon dioxide (a product of the combustion) build-up alert system and not "lack of oxygen in the body" alert (unless you consider oxygen deprivation symptoms as such). So, my question is: how come people don't wake up when being poisoned by carbon monoxide if carbon dioxide is a main product and in an environment when oxygen is insufficient for carbon monoxide to be produced, carbon dioxide would still be in the space and not vented out?
Part of the resolution of your question rests on the definition of "insufficient oxygen". Part of the question relates to the relative threshold for oxygen deprivation and CO poisoning.
A chemist using the term "insufficient oxygen" is referring to the process of combustion. You appear to assume it applies to the whole room. In real combustion reactions, it is very easy to have insufficient oxygen in part of the flame even when there is a good supply of oxygen to the fuel being burned. Even well regulated heaters based on burning fuel often produce some CO even when the carbon dioxide dominates. This is why regulations insist on good ventilation of the exhaust gases from heaters.
The scenario you describe is neither common nor relevant to worries about CO. If a flame used up all the oxygen in a room and replaced it with carbon dioxide, you would surely be asphyxiated, but you would also notice very early that the carbon dioxide level was worrying long before that point.
But what happens with badly ventilated heaters is not that. Leakage of CO into the room is the dominant problem not a lack of oxygen. CO is very poisonous because it forms a relatively stable compound with haemoglobin, blocking the blood's ability to move oxygen around the body. This happens at very low levels of CO, long before any external shortage of oxygen has happened. When town gas supplies consisted of hydrogen/CO mixtures (phased out in favour of natural gas which is mostly methane in the 1960s in the UK) "putting you head in the oven" was a favoured method of suicide. As was piping a car exhaust into the car (which became harder post catalytic converters which remove CO and some other gases).
The point is that a small level of CO in a space where there is plenty of oxygen is deadly. And the deadly levels are far lower that the levels of carbon dioxide that would trigger any physiological warnings in the body.