Well, you’re sort of correct.
In Festival Plaza, there are 3 different lottery shops: Big Dreams, Gold Rush, and Treasure Hunt. Each of the 3 lotteries have 10 different items to win. The 3rd-10th item to win is the same for all three booths, but the 1st and 2nd prizes vary based on what shop you have. Also, the rank of the booth does play an effect - where in a 1-star rank, you may find yourself getting a lot of Berry Juice and Full Heal, but in a 5-star rank, it's much easier to win the top prizes.
So, to answer your question, the different lottery booths don't effect your outcome, but instead the types of prizes you win. The rank of the booth is what adjusts the odds, where 1-star and 5-star being the lowest and highest rank respectively. You may refer to the this or this link if you need more clarification, as well as seeing the prizes they have to offer.
Hope I helped! :)