TL;DR: There is a basis of fact in claims that sex existed in the geisha world. However, it is not always the case.
Geishas are trained in a variety of art forms ("Geisha" literally means "Artist") that prostitutes would not be trained in. So in that sense, geishas are not prostitutes.
But is sex a part of their work?
Wikipedia is a tertiary source, often edited by people with widely diverging views, who insert claims that fit into their opinions on the subject. As a result, some parts of an article are not entirely consistent with other parts.
However, in the case of geishas, the variety of claims is not the result of Wikipedia having a poor editorial process. Even secondary sources written by a single author, such as Liza Dalby's "Geisha", say that it is a complex issue.
In Geisha, Dalby mentions in a number of places a variety of factors that can influence whether sex is involved in being a geisha. This includes the time period you are examining, whether you're looking at geishas who live in major cities versus onsen town geishas, and even which geishas districts you are in for a given city.
It's stated that the role of sex varies between geisha districts (hanamachi) on page 6:
Midori's natural mother was a retired geisha in the nearby area of
Miyagawa-chō, one of the six recognized geisha communities
(hanamachi) in Kyoto. Why didn't Midori become a geisha there? I
wondered. This was the sort of thing one could not inquire about
directly but about which my okāsan, ex-geisha, mistress of an elegant
inn, and pillar of the Pontochō community, could enlighten me as we
had tea in the afternoon or snacks late at night after a party at her
establishment, the Mitsuba.
"You're making a study of this, Kikuko," she said, using my ordinary
and familiar Japanese name, "so you should know about Miyagawa-chō.
There's a word called 'double registration' - that's what many of the
geisha in areas like that are. You can call them geisha if you like,
but they do a bit more than dance for the customers."
[Notes section] "Double registration" (nimae kansatsu) dates from the
days when prostitution was legal in Japan. Prostitutes had to be
licensed as such, and geisha were licensed as geisha. A woman could
not hold both licenses, so double registration became the derogatory
description for a geisha who slept around.
It's stated that the role of sex has varied over time. On page 116:
Sex undoubtedly used to be simpler in the geisha world. A maiko was a
maiden, and her sexual initiation was part of becoming a full geisha.
Ordinary girls put aside their swinging sleeve kimonos when they
married; maiko put aside theirs when they graduated into geisha. In
both cases, adulthood presumed sexual experience. A virgin geisha
would have been as odd as a virgin wife. Now, however, freedom of
choice has muddled these once straightforward categories. I know a
maiko who has stolen away to meet her young boyfriend in one of the
hotels that offer special day rates. On the other hand, Ichiume
and Ichiteru passed into geishahood with maidenheads intact.
Geisha generally know more about sex than housewives do, but a man who
thinks of a geisha's gei [art] as rampant eroticism will be
disappointed. Even in the long-gone era of licensed [pleasure]
quarters, geisha were not the foremost sexual adepts. The appeal of a
romantic entanglement with a geisha has always embraced more than sex.
[Words in square brackets added]
On pages leading up to the following quote, there was the description of the host of a geisha event helpfully describing mizuage to Dalby. Dalby's matter-of-fact description of how older geishas feel about the topic makes it seem like someone describing reality, rather than feeding the reader's preconceptions back to them:
"What about now?" I asked, seeing this as a chance to find out more
about sex in the geisha world - a topic that geisha are understandably
touchy about. "It's all changed now," said the okaasan. "There's no
mizu-age ceremony any more, with or without eggs. All the maiko have
been through junior high school, so they aren't as ignorant as we were
- right, Ichiume? They pretty much pick their own boyfriends and patrons when they're ready. That's not the same as mizu-age."
I sensed a faint embarrassment in a few of the older geisha as the
curious young ones listened. Today geisha and women in general have
more control over their sexuality. Older geisha automatically say how
wonderful it is that their daughters do not have to submit to
mizu-age. Yet this means that their own experiences, far from being a
useful guide to the younger women, are dismissed as "feudalistic" - a
term used in Japanese not only in a political sense, but also to refer
to any practice considered passe, unenlightened, or simply
unfashionable. I often found that the older geisha spoke more freely
about sex when the young women were not present.
Liza Dalby's okāsan was a genuine geisha. She wasn't a prostitute who put on a kimono and pretended to be a geisha. She's as real as real can get, and probably as prestigious and up-market as any.
On page 173, Dalby talks about onsen geisha. To answer the question "Are they still considered geisha?", Dalby says they are, but that combining different categories of geisha is ludicrous. Before the quoted section, she talks about some derogatory terms for geishas, including "Korobi" (roll-over) geisha and "Daruma" geisha (named after Daruma dolls, which go horizontal very easily).
The more I found out about the geisha world, literally the world of
flowers and willows, the more diverse it appeared. But soon I
understood one thing. The common misunderstandings and arguments about
the connection between geisha and prostitution springs from an
indiscriminate collapsing of categories of geisha. Whereas in one
sense we may speak of a Kyoto apprentice and an onsen geisha in the
same breath as part of Japan's living geisha tradition, in another
sense combining them at all is ludicrous. When I began my detailed
look into the various characteristics that differentiate geisha - the
part of the country where they work, the prestige of their particular
hamamachi, their ages and so on - it became clear that I was dealing
with a very complicated phenomenon indeed, and that making an easy
general statement about "the geisha" would inevitably be misleading.
When stating that variations exist over time, or between different geisha districts, my intention is not to say that sex always exists in some circumstances, and never in others, but merely that it is a complex issue.
The book "Autobiography of a Geisha" (not related to the book "Memoirs of a Geisha") provides a first-hand account of someone who was a geisha for part of her life. She had to do sex as part of being a geisha. I have not read the book yet, but a couple of reviews of the book note that she underwent mizuage a total of five times, even though it's supposed to only happen once. This review notes that she learnt several geisha arts, such as dance, song, shamisen, and song, so she wasn't just a prostitute pretending to be a geisha.
As a counter-example based on personal experience, I had dinner with two geishas, plus a translator, in Aizuwakamatsu, Fukushima prefecture. They did not have sex with me. Instead, the geishas performed two dances (one for me, a guy, and one for my translator, who was female), played a variety of ozashiki asobi (geisha games) with me and the translator, and talked a lot. A further description is available on Travel Stack Exchange, and on lang-8.com.