Welcome to the Village. Many have given you the basics on the primary saws to consider.As Ray, Octoolguy, noted to help you zero in on the best saw for your budget, we would need to know what the budget is. What one person can afford to spend on a saw, versus another can vary greatly. Also, as also pointed out, how frequently and how much daily you think you are going to use it can also impact recommendations. Many of us are hobiest and don't cut everyday, or don't cut for hours and hours on end. Others do cut everyday and for hours at a time. The saw that works for each of those may not be the same.
Finally, since you mention you want to get one to start to cut "puzzles" there are some key things you need to learn to do to make sure you are cutting a puzzle that can be put back together. Couple of keys is that you want to use a blade that doesn't remove too much kerf as Dan pointed out. This means the thinner and smaller the blade the better. However, you also need to make sure you are cutting square which means among other things, you are not "pushing" the wood into the blade, but letting the blade cut the wood. You will also want to make sure that you are properly tensioning the blade. Any flex to the blade and you will not be able to cut the pieces square. Being square is extremely important in cutting puzzles because if you are not, you will find you cannot take the puzzle apart and/or put it back together. Pieces will not fit and slide back in forth in both directions.
The puzzles you indicated you are looking to cut, you said almost solely jigsaw puzzles under 1/2" thick - but you didn't say exactly what type of puzzles beyond that. There are lots of different types of puzzles cut with a scroll saw. Some are true, thin picture style puzzles that lay flat and have a few or many pieces while others are thicker "object" puzzles like animal puzzles like @Iguanadon Iggy (Scott) cuts and sells. Knowing what type of puzzles you are wanting to cut can also help people give you the bst help and advise.