If I am "on a budget," it means that I am trying to save money and buy something cheaper than usual. "I am looking for a laptop, but I am on a budget" or "I am looking for a budget laptop" means that you are prioritizing a cheaper laptop. Search online for "budget laptop" and you can see some articles that give good context.

I have a limited budget. I don't have extra money to spend.

I was on a budget so I rented the smallest room possible.