Another pic showing the overall door trim profile might help but the close up is valuable. From the little section of the wall that I see, if it were my house I would razor knife the white paint at the point where the trim meets the wall. If it falls away.... good. If it doesn't, maybe the next step may clear it off. Which is to remove the raised line of wall paint that will still be seen through the new coat of paint and it will get in the way of a new line of caulk which will be done in the way of prepping.
To remove the raised line of paint is to wrap a 1/4 or 1/6 sheet of 100 grit sandpaper around a block of wood and sand the inside corner where the knife cut was made. This will flatten the paint line of wall paint and perhaps clear the left over trim paint chips that did not fall out at the time of cutting it out.
After all the lines are flattened, and the inside corner is clean of loose debris, you can prime it if you like to get an idea how it will look after it is finish painted, and to give a base to caulk the inside corner, that will fill the joint between the wall and the wood trim.
Here is a tip for caulking- I always have a 5 gallon bucket filled halfway with water and a round corner sponge, like tile setters use when gluing or caulking. Any mistakes or excessive caulk or glue can be readily cleaned up with the sponge that is well rung out. Using a smaller bucket filled will have water everywhere when the sponge is squeezed, at least for me there was, so half way for me it is, that way I keep the sponge well below the top of the bucket and squeeze away, no water flying everywhere....