Roman bench, side bends on roman bench, crunches, I do incline situps but crunches are the safest way.
The last picture, you can do many varients of this, up incline, down incline, and many other cable exercises also work core, abs for down, back for up, obliques for up side, lats for down side but the main benefits come from roman bench.
Then you do pullups, pulldowns for back traction, in line with body.
All these are important. The back pain should decrease in about 3 months after every other day exercises. I bought my own roman bench. This sounds painfully slow because it is. Would you rather have slipped discs? Nerve pinches? Arthritis? Bone spurs? That's your future without core after 30. I had three bulging discs, Arthritis, scoliosis, and a pinched nerve presenting as discomfort in certain sitting positions, that progressed to a "60 yr-old's back". I couldn't get up from a chair, roll over in bed, sneeze etc without intense pain and stiffness, or bend certain angles at all. Forget about lifting anything.
They told me I'd need surgery, medication for the rest of my life, and that still only had a 50% chance of improving symptoms.
I went to three different doctors before someone said physical therapy and I had to hope it would do something because it did nothing for weeks, but at the 2 month mark I saw some marginal improvement and by the three month mark I saw significant improvement and when they took a follow-up xray the doctor that said I needed medication for the rest of my life, after lecturing me for not taking the medication dispite the improvements in symptoms, saw the new xrays and said, "that's impossible" because according to her you can't solve any of my conditions with exercise. In fact she said exercises would only make it worse and at twentysomething I should really just have surgery and take my chances.
No fucking way am I flipping a coin to have permanent bone fused bullshit.
A year later, I only had pain after working too hard, after three I only had pain if I stopped doing the PT, now after many years I don't do the PT more than once or twice a week and I haven't had any back pain for years.
As a measure of core you do planks. I did 600 seconds because the master at Ju-jitsu said he did and so I did and could keep going but it's boring af and what was it even doing except make me drenched in sweat.
So if you want a healthy back you need to work for it. There's no other way and you need to do this before you get that slipped disc because that is probably not something that exercises can fix.
