I've had Budgies, cockatiels and a cockatoo, and it's not true that Budgies are not that intelligent! The world record talking bird is (or at least was) a male Budgie that "spoke" over 1700 words! Budgies are VERY intelligent, can be trained, and bond closely to their humans. Plus they are affordable, beautiful, playful, not too loud and messy, and independent enough not to suffer excessively from separation anxiety. They very rarely pluck their own feathers out when not given enough attention, but they do need lots of attention to thrive, and they aren't potentially dangerous like some large parrots that can inflict a pretty substantial bite. Budgies don't require an enormous expensive cage- 30x18x24 is perfect for one Budgie. Only drawback with Budgies is that once they outgrow babyhood at six months or so most of them do not like to be petted or cuddled even when they were hand raised.
I don't think you can get a grey, an eclectus, or an Indian ringneck for $1,500 unless you adopt an older bird which may have behavioral problems that may be difficult to deal with if you lack experience with parrots. A hand raised fully weaned "baby" is ideal for a new parrot owner, but baby greys, eclectus, and ringnecks usually cost several thousand dollars, not including a 36x36x36 cage which runs at least $500 new. Also consider how long a larger parrot can live (30+ years is typical). If you are over 50 your bird may outlive you requiring you make arrangements for it's care after you pass. A healthy Budgie can live 10 to 15 years which may be better for you.
A hand fed baby green cheeked conure or sun conure would be an excellent midsized parrot for someone craving the "big bird" experience. They make amazing companion birds and for $1,500 you can get yourself a hand fed baby plus the 36x24x24 cage they require. Hand fed cockatiels are also delightful larger small parrots and are well within your budget for a baby. They bond very strongly to their humans and although they aren't great talkers they can whistle entire songs, especially the males. They can retain their love of head scratches and being petted more than Budgies usually do, and they are very easy going, affectionate, and not too loud and messy. They can live over 20 years.
I don't recommend a cockatoo. I had a Goffin's Cockatoo and he was emotionally extremely high maintenance and not in a good way.
I don't know much about lovebirds so I can't say.
I hope you find the right bird for you. Take your time because adopting a bird is a huge commitment and you want both you and your bird to have many happy years together!