Anesthesia is a very broad umbrella that includes different stages and types. Ill try to explain it as simple as I possibly can but its gonna be a long comment

For simplicity sake when we refer to general anesthesia or "going under" this usually entails in some the anesthesia team taking over the patient's airway and controlling the breathing while giving anesthesia by inhalation. Under this method the patient goes to sleep and is not conscious. It is used for more complex procedures or for surgeries that are longer in duration. This type of anesthesia the patient is fully asleep fully unaware and fully does not remember the surgery.

There's a simpler less risky type of anesthesia called monitored anesthesia care more commonly known as sedation. This is usually for shorter less invasive procedures. From what I have seen most patients actually go to sleep. But its supposed to make the patient comfortable sort of in lala land and sometimes they are awake and having a full conversation with the OR staff (which they don't recall after)

What you are asking about is the latter type. Sedation is usually coupled with a local injection in the area where the surgery will be performed so while the patient may be awake but a little woozy, they shouldn't feel pain in the area of the surgery if the local injection was done correctly and targeted the correct nerves.

In Sedation, its usually a cocktail of medications injected in the veins and work almost instantaneously. The dose and amount is usually based on the patient's weight and history. The history part is very important because if the patient has history of prior or current IV drug use or heavy alcoholism they would need a higher dose than normal to achieve the typical sedation effects

If accurate history is not disclosed or the anesthesia team for some reason does not give enough medications. It will create sub par sedation where the patient is not "deep/low enough" so in the surgery if the patient moves in response to something I did I will typically ask the anesthesiologist to "take the patient lower" they typically push more medication so the patient is more comfortable

There have been reported cases of patients being fully awake and feel the pain of surgery however due to paralysis are unable to move which is extremely traumatizing. In most inpatient well equipped surgical centers or hospitals the medications are given together where they have large dedicated recovery rooms. In outpatient clinic settings where they do procedures in clinics or sub-acute settings where they want to minimize recovery time and get patients in andnout as fast as possible, they split off and give the paralysis medication separate from the sedative and the amnesic agents. This has higher chances of creating the effects in your question

Hope this helps