Pink swollen eyelid, sore throat/cold. Dr said stye, but there is no bump at all! Just an overall swelling/pain. Incorrect diagnosis perhaps?

Probably right. Most styes start out with diffuse swelling of a part or sometimes the whole lid. As your body starts to deal with it, it becomes localized to the more typical focal redness and swelling. I hope you were told to apply heat 4-5 times per day which is in face the best treatment.