Activating core is something you want to do as much as safely possible. You recently had someone forcing your entire abdominal wall outward slowly over hopefully 9 months, who then exited and left you to sort out the damage.
You are now a year out so the joints will have tightened back up and unless there were complications you will be safe to train as anyone else would of your fitness level. If there were complications check with doctor to find out what your limitations are.
Next the bad news. It took 9 months to have the muscles stretched out to their furthest position, it will take longer to put them back, do not be in a rush, because this will likely lead to injury.
General advice.
Try not to think of your core as a group of separate muscles to work one at a time. The core contains some of the most commonly damaged postural muscles, unbalanced training here will almost certainly lead to back issues.
Train to the shape you want or need. I have a strong core, it doesn't stick out quite as much as yours but is far from flat. I have built this because it is useful to my lifting. If you want to have a flat stomach you need to train and live with this in mind. I had this and let it go by deciding it was more important to be able to do more than look pretty. Ironically the time I started getting the best defined and flattest stomach was when I stopped thinking abs and went predominantly compound, so my core was being worked regularly.
Training, sort of.
First and foremost the core is postural so if you want it flat, pull it flat. If you are anything like me you will expect this to be nonsense, I did when told, but it does work better than any short workout session. When standing don't let your core drop, pull in and up and keep it there, at first you will not be able to do this full time but after some while it will just become part of your normal stance. the same comes when sitting, there is a back on my work chair, it is virtually never used, I keep myself upright and tight most of the day even now. The pull is up from the bottom and in further up, and you should never simply release, when you tire relax a bit but don't flop. This becomes most difficult when pushing a chair or pram, a time when some of the worst posture you will see becomes very apparent, keep upright and tight while walking, your body will build to the shape you give it. These are postural muscles so can withstand working all of the time in this way. All the training in the world will not flatten your stomach as well as this I will promise you. There is also not a workout on this planet that requires as much commitment either so you earn the flat torso.
When you are training for core ironically you want to work everything, the best way to do this is using free weights, they don't have to be heavy but the stability required will hit your core on every movement. If these aren't available to you use bodyweight training instead. There are too many myths about lifting iron making you look like Arnie etc. but the reality is very different, consider iron training is the preferred for every bikini model worthy of note because they know this is the only way to get the toned look needed when displaying so much flesh.
The key is to keep the tight core during all exercises. So if you are doing a deadlift you get into the get set position, pull the core tight and hold it tight throughout the entire movement. The most intense core activation I have felt from any exercise is from overhead squats, holding a bar over my head at arms length squatting deep and coming back up under control, the weight I was using was pathetic in my world but I would say it would have still my core well if I had done high reps using a broomstick. Doing this exercise makes it easy to let the core bloat out, I don't mind doing this because I am not wanting a flat stomach, you will have to resist this and hold it in, trust me you will feel this.
Other movements include standard squats, arched bench press, high pulls, any standing weight training in fact will help you if you fully engage the core and pull it up and in. Remember it's not about the weight, you could be using a broomstick or empty bar the important part is the technique and focus.
Cardio work. Same rules apply, if you are on a rowing machine, cross trainer, treadmill, pull tight in and up, this will be harder the higher the intensity, but you have to keep focussed. This will cause issues when needing to breathe deeply so you may have to take slightly shallower breathes in the short term to get what you want then expand your breathing range gradually, if you can simply keep tight through deep breathes from the start this will be better but if in doubt err on the side of caution.
I haven't been pregnant, the name may have given this away, but the system I am recommending here is what worked for me when I wanted my stomach to be a pretty flat 6 pack. I had this for a number of years. Now I can still pull a 6 with little effort but because I have built my core to be bigger and stronger, it is a protruding six rather than the flat aesthetic trainers style.
