In 2013 I had a sty on my eye. It went away but left my eyelid looking lazy. Is this ptosis or just a droopy eyelid?

There is definite left upper eyelid ptosis, which means droopy eyelid. Long-standing styes or chalazions can cause the eyelid to get droopy.  You may be a candidate for ptosis surgery. See an oculoplastic surgeon.

It sometimes happens that inflammation can be the cause of an upper eyelid ptosis.  The swelling that is related to the inflammation probably causes some damage to the tissues that hold the eyelid open.  A careful consultation with a fellowship trained oculoplastic surgeon will determine if you are a candidate for ptosis surgery to restore symmetry to your appearance.  There are two principle methods of repairing upper eyelid ptosis and during the consultation it is possible to determine which of these surgeries would be right for you.

This certainly looks like a droopy eyelid which is called ptosis.  A stye can cause this rarely.  You would need a consultation with an Oculoplastic surgeon to decide which procedure is best for your eyelid.  The cost would depend on the procedure  As for your last question, eyelid lift usually describes a blepharoplasty which removes excess skin and/or fat.  This would not help correct true ptosis.