Is there such thing as allergic conjunctivitis and not bacterial conjunctivitis?

Allergic conjunctivi: Yes, there is definitely a condition called allergic or vernal conjunctivitis which is not contagious unless secondarily infected by rubbing the eyes. It is usually accompanied by sneezing, stuffy/runny nose and possible a cough.

Absolutely!: Allergic conjunctivitis is very common, especially now in spring. Itching, red, burning associated with pollen, animals, dust, mold. Responds to allergy eyedrops, antihistamine pills. Viral conj. (pinkeye) also common, very contagious. Bacterial eyedrops, often required by administrators for return to school for red eyed kids, do nothing for these; only the rare case of bacterial conjunctivitis.