There is no specific, correct way to train and lose weight other than follow a weight lifting program and eat at a deficit. The deficit is what will cause the weight loss. Any fitness, weight training included, can help some but is more for other goals, like getting stronger. Any program geared towards strength will help in that regard though you may see an earlier slow down in the strength increases while eating at a deficit compared to maintenance or above since fuel helps, but you can definitely get stronger while losing weight.

Three weeks isn't very long and have to remember there are a fair number of factors that affect weight. I know some women see a 5 lb increase in just one week during their monthly cycle. Hormones are just one factor as hydration levels, sodium intake and muscle repair can cause slight increases as different times. And I would think if there is more visible muscle definition then that is an indication of fat changes because losing a little body fat will help get that definition.

Some tips in general:

1. Make sure you're able to be as accurate as possible with the calorie intake. Food scales are recommended, especially if you're not having a lot of weight to lose as the margin for error on the smaller deficit is minimal.
2. Careful on the numbers for exercise. Do need to eat back some of the calories burned from exercise unless following TDEE (as activity is factored in to get the numbers already for that method) but it's not always easy to get accurate numbers.
3. Give it some more time. Let your body adjust to the activities a little and keep on pushing.
4. Take other measurements. Measure waist, hips, arms, etc and keep track. I do once a month measuring both know others who do it once a week.
5. Try to weigh in at the same time of day and with similar conditions (same clothes or lack thereof, etc). I tend to always weigh in the morning but I know some weigh themselves at the gym at night. Long as you are consistent the changes should be noticeable.
6. Remember weight loss isn't linear. It has ups and downs. It takes time but use that to enjoy the process. Have fun lifting and focus on more than just the numbers on the scale as that will make things easier.

Thanks for the info and support I jumped in to read the ideas!!!
Good luck newbie... you can do it.