Jogging for 20 minutes three days a week will not help you lose fat and look slimmer. In fact, it will start to slow your metabolism down as your body adjusts to the work, which will actually make you start to gain weight.
Just a heads up, this is a website dedicated to weightlifting. It's a little weird to create an account here if you don't lift weights. ("Do U ev3n lift BRAh?) This is not even to mention the fact that the best way to lose weight is a combination of compound liftng and intense conditioning.
The calorie calculator you found online seems accurate. You will have to experiment over time to find the caloric deficit that works best for you. More likely than not, it will be closer to 500 than 100-150. Again, YMMV.
I think a strength and recomposition is a good call. What this means (please correct me if I'm wrong, T-Nation experts) is that you can think about gradually building muscle and dropping fat, to change the ratio of lean mass to fatty mass in your body overall. However, it is hard to say for sure if a traditional cut or a recomp would be better, because we're not sure how "bad" your situation is. Bodyfat percentage, pictures, and diet information would be helpful. Blur your face out of the pictures if you want.
In the meantime, you need to start resistance training. Barbell training is best, and for that I recommend Stronglifts 5x5 and Mark Rippetoe's Starting Strength programs. I dare say almost everybody on this forum started on a program just like those two. If you can't start barbell training for a while, body weight exercises will do in the meantime.
To get you started, do this three times a week:
Push-ups: 3 sets 1-20 reps
Pull-ups: 3 sets 1-5 reps*
Body weight Squats: 3 sets of 1-20 reps
Plank Hold: 1 minute
Sprints: 20 minutes**
Do as many reps as possible for all three exercises.
*If you can't do pull ups, then do "negative pull-ups": use a box or chair to jump up to the top position of a pull up. Hold for a moment, and then lower yourself slowly. Then use the box to jump back up to the top.
**For these, pick a length of street. Warm up and stretch. Then, sprint as fast as you can from one telephone pole to the next. Then lightly jog to the next telephone pole. Then sprint to the next one. Carry on in this manner for 20 minutes, or until you throw up. (Joke)
As a general note, you need to show some fucking hustle. Nobody is going to hand you a better body. Even a personal trainer can't give you what you want. Only you, and your learning, and your motivation, and your dedication, and your discipline will result in you being happier with your body.