Is a stye an internal hordeolum?

Sometimes: A stye is the common term for a pimple like inflammation of the lid and lid margin. If located on the edge it is termed an external hordeolum and if in the substance of the lid, it is termed internal hordeolum. Your ophthalmologist can recommend the best treatment for this.

Yes and more: Stye is a general non-medical term used by non-ophthalmologist describing generally any pimple like lesion of the eyelid. Most people do interchangeably use stye for hordeolum and chalazions. Hordeolum and chalazion, are same except hordeolum usually appears red and painful, suggesting an infection. Chalazion are not red or very painful. Either can be internal or external.

Clogged oil gland: A hordeolum (often referred to as a stye) is when an oil gland (mebomian gland) of the eyelid margin becomes clogged. The oil backs up and forms a nodule and inflammation. An internal hordeolum is when the nodule protrudes on the anterior surface of the eyelid and a internal is when it protrudes on the back surface of the eyelid.