I've seen in several places recommendations about caulking around a window frame to reduce heat loss. I have cracked caulk around one window where you can feel air flowing in. I was going to fix it, but does the airflow mean there's actually a crack in the caulking on the outside? If I just re-caulk the inside, won't that air just end up in the walls or should the framing around the window be sealed off from the walls? Is it okay to re-apply the interior caulk and then look at fixing the exterior caulking when it's warmer outside (currently way too cold to apply caulking outside).
Caulk is usually applied only outside. That's where airflow should be addressed. Caulk on the interior makes for an unappealing appearance and, as you suggested, treats a symptom and not the primary issue.
I would apply interior window film, if you think the remaining winter season warrants it, and deal with your window installation issues in warmer weather. Windows installed in the last few years should've been foamed in, which creates a fantastic seal to the framing. they should also be caulked to the housewrap or wall sheathing as the situation allows. No air management should need to happen on the inner face. Window trim is not intended to be a component of the building envelope.
Those look like vinyl replacement windows and it looks like the trim has separated away from the window or there was always a gap when it was installed. I've always seen caulk used on the interior of these types of replacement windows around me.
You can caulk the inside and deal with the outside later when it's warmer. For such a big gap you may want to use some low expanding window & door spray foam first. You'll want to do the exterior soon though because where air can get in so can water which will cause worse issues than just a draft.
Typically the air won't go anywhere else in these types of retrofits unless there was a sash weight and the hole didn't get plugged up with insulation.