Can you have a carcinoid tumor on the lung, not in the lung?

Yes: I am guessing from your question, but yes there can be pulmonary and bronchial carcinoid tumors. They are both a type of lung tumor that can be benign or cancerous. They can occur in the lung or "on" the lung in the breathing tube to the lung. These are relatively rare tumors.

Yes: Carcinoids are cancers of Kulchitsky cell origin. In the chest they can be in the lung tissue, or arise from the walls of the tracheobronchial tree or in the thymic gland the last 2 of which could be considered "on" the lung. Typical carcinoids represent one end of the spectrum of lung cancers (more indolent) but another cancer derived from Kulchitsky cells - small cell cancers- are the deadliest