Published: August 15, 2020
Revised: June 23, 2022
The dictionary defines an organ as "a differentiated structure (such as a heart, kidney, leaf, or stem) consisting of cells and tissues and performing some specific function. An alternate definition is "bodily parts performing a function or cooperating in an activity." Merriam-Webster gives "the eyes and related structures that make up the visual organs" as an example. 
The penis is the most visible organ in the reproductive and urinary systems. And the penis is the primary organ involved in male sexual pleasure. The glans and the shaft (to which the foreskin is attached on an uncircumcised penis) do not constitute separate organs - any more than the pupil or retina of an eye would constitute a distinct organ.
Some have referred to the foreskin itself as an organ,   apparently because the body's skin structure is classified as an organ. In fact skin is considered the largest organ. Some have referred to the glans as an organ.
While the entire skin structure constitutes a protective organ, no fractional portion constitutes a distinct organ. Most scientists accept this definition instinctively. Researchers in Europe and Asia report that circumcision can be used to treat various medical conditions while preserving the organ.  
External and Internal Organs Several intactivists have described the glans - or the entire penis - as an internal organ. [6-8] They say that removal of the foreskin changes the glans or penis into an external organ. These claims signify a misunderstanding of the terms.
An external organ is an organ located on or near the surface of the body. An internal organ is an organ located inside the body. Urologist Dr. Rena Malik explained, "The male reproductive system is mostly located outside of the body. These external organs include the penis, scrotum, and testicles. Internal organs include the vas deferens, prostate, and urethra." 
Having a skin covering doesn't transform an external organ into an internal organ. The testicles are classified as external organs.  During his entire life a man will never see his testicles, which are permanently enclosed inside the scrotum. Yet the testicles are classified as external organs - because they are position outside the body - external to the body surface. If testicles - which are never visible - are external organs, then even moreso the glans - which is outside the body and visible during urination and sexual activity - is (part of) an external organ.
The only time that the penis is an internal organ ... is during sexual intercourse.
 For example: Georganne Chapin; "Lousy Sex"; Intact America Blog; April 8, 2014. "Without a foreskin and its sensory feedback, a man has difficulty controlling the timing of his orgasm. Also, because he’s missing the very organ that serves a gliding and lubricating function ... his penis is calloused and dry, when compared to that of an [uncircumcised] man; this creates a friction during intercourse and compromises the pleasure of both sexual partners."
 "Circumcision to Reduce Men's Sexual Pleasure"; Saving Our Sons; July 2012. "The prepuce ('foreskin') was well known to be the key organ at play in whole sexual health..."
 Gregory Nason, Clare O’Connell, Paul Hegarty; "Organ-Preserving Surgical Treatment of Penile Cancer and Glans Lichen Sclerosis"; Textbook of Male Genitourethral Reconstruction, pp 727-735; November 13, 2019. "Primarily, patients with lichen sclerosis can be treated with topical agents and circumcision."
 T.B. Yuvaraja et al; "Organ Preservation Surgery for Carcinoma Penis"; Indian Journal of Surgical Oncology; March 2017. "This review focuses only on the penile-preserving surgeries which would help to understand the options available if one opts for a penile conservation surgery... Circumcision: Uncircumcised penis is a harbinger of malignant change occasionally. Being the most common surgical procedure in the management of penile carcinoma, [circumcision] is indicated for symptomatic treatment of painful or hemorrhagic tumors as well as for acquired phimosis secondary to preputial tumors. Also, it is always recommended before radiotherapy as it allows better exposure, targeting, and definition of the tumor."
 "Does being born a healthy male require surgical correction?"; Doctors Opposing Circumcision; uploaded January 2016. "The glans is a mucoid internal organ, like an eyeball."
 Paul Tinari tweet: November 12, 2019 - 12:26 AM. "The penis is an internal organ with bacteria that live on mucous membranes compatible with bacteria that populate the vagina. Circ changes the penis into an external organ..." NOTE: As of April 2022 Tinari had limited access to his tweets.
 Gallus; "4 Stupid Reasons Why People Circumcise Their Children"; Foreskar; September 3, 2018. " The penis is an internal organ, that’s why it has a foreskin."
 Rena Malik MD; Facebook post; June 22, 2022
 "Male Reproductive System"; Cleveland Clinic; last reviewed November 7, 2016. "The male reproductive system is mostly located outside of the body. These external organs include the penis, scrotum and testicles."
 "The Male Reproductive System"; Lumen. "External Male Sex Organs: Most of the male reproductive system is located outside of the man’s body. These external structures are the penis, scrotum, epididymis, and testes."