No. An internal hordeoulm is the acute clogging of an oil gland in the eyelid margin (mebomian gland), it is not an infection. It is usually red, sore, and swollen. A chalazion is the chronic form of a hordeoulm, but the tissue becomes granulated, and is not red, painful, and inflamed, but there is still a nodule or bump.
Yes, sort of. The difference is location of infection being on the conjunctival side (not outside). Needs treatment of antibiotics and drainage/removal. See your ophthalmologist for care do not ignore. They are the same process.