Ah, sorry. What I meant is, with Turkish you will often see prices in a booking class that are lower than what you can actually book on Turkish's website. And then it's the Turkish website that is correct.
Not sure what exactly is happening in your case. Budgetair should be safe, but of course with all of these cheap OTAs, it can get messy if you need any kind of customer service, they may have fees for rebookings/refunds etc.