First of all, as a disclaimer, it does happen that experts will disagree with each other and that no consensus is reached on what the definitive answer is. That being said, this is typically because the available evidence can be variably interpreted and used to argue for competing theories, none of which can then be taken as conclusive. It may be that conclusive evidence has not yet been found or it may be that no such evidence can exist.
In the absence of any detailed arguments presented in support of either theory and upon consideration of various evidence, which I will outline below, I will argue in favour of theory 1, that kawazu-gake is named after the famed sumo wrestler, Kawazu Saburo Sukeyasu and that the image of a frog applying a gake was for humorous effect or similar, because “河津” (Kawazu surname) and “蛙” (frog) are homophones.
The Japanese Language
Chinese characters first came to Japan on official seals, letters,
swords, coins, mirrors, and other decorative items imported from
China.
The Japanese language had no written form at the time Chinese
characters were introduced, and texts were written and read only in
Chinese. Later, during the Heian period (794–1185), however, a system
known as kanbun emerged, which involved using Chinese text with
diacritical marks to allow Japanese speakers to restructure and read
Chinese sentences, by changing word order and adding particles and
verb endings, in accordance with the rules of Japanese grammar.
Chinese characters also came to be used to write Japanese words,
resulting in the modern kana syllabaries. …the two other writing
systems, hiragana and katakana, referred to collectively as kana, are
descended from kanji.
In modern Japanese, kanji are used to write parts of the language
(usually content words) such as nouns, adjective stems, and verb
stems, while hiragana are used to write inflected verb and adjective
endings and as phonetic complements to disambiguate readings
(okurigana), particles, and miscellaneous words which have no kanji or
whose kanji is considered obscure or too difficult to read or
remember. Katakana are mostly used for representing onomatopoeia,
non-Japanese loanwords (except those borrowed from ancient Chinese),
the names of plants and animals (with exceptions), and for emphasis on
certain words.
https://en.wikipedia.org/wiki/Kanji#History
Kanji
Kanji (漢字, pronounced [kaɲdʑi]) are a set of logographic characters
from Chinese script which forms a major part of the Japanese writing
system alongside with Japanese syllabic scripts hiragana and katakana.
The Japanese term kanji for the Chinese characters literally means
"Han characters".
https://www.fluentu.com/blog/japanese/onyomi-and-kunyomi/
Remember, while Japanese didn’t have a written language, it did, of
course, have a spoken language prior to the arrival of Chinese
characters, and the attempts to merge the two helped to create these
natural variations in kanji readings.
https://en.wikipedia.org/wiki/Kanji
So, when a Japanese person reads kanji, how do they then pronounce the character? Do they pronounce it as it reads in Chinese, or do they take the meaning of the Chinese character and use the native Japanese word?
The answer is both. Kanji can be read in different ways, but the two main ones are: kun’yomi and on’yomi.
Kun'yomi
Kunyomi is what’s known as the fully Japanese version of the kanji
reading. It’s the way that the Japanese culture managed to merge the
kanji with the existing spoken language they had prior to the written
one. In this circumstance, the Chinese meaning of the original
character is kept, but the closest equivalent to the word in Japan’s
spoken language is associated with it. As you can imagine, these words
weren’t always a one-to-one match, and the Japanese language added its
own connotations as needed. Hiragana was later added to kanji,
tweaking readings and bringing more meaning to each kanji they’re
associated with.
Kunyomi is a near guarantee when you’re using proper nouns. Most
notably, it’s used when you’re using native Japanese proper names.
Most of the common family names are pronounced using the kunyomi
version of the kanji.
https://www.fluentu.com/blog/japanese/onyomi-and-kunyomi/
The kun'yomi (訓読み, [kɯɰ̃jomi], lit. "meaning reading"), the native
reading, is a reading based on the pronunciation of a native Japanese
word, or yamato kotoba, that closely approximated the meaning of the
Chinese character when it was introduced.
https://en.wikipedia.org/wiki/Kanji#Kun'yomi_(native_reading)
On'yomi
Onyomi translates roughly to “sound reading.” It means that the kanji
is read the same way that it would in the Chinese language—or at least
is read as a close approximation of the sounds that are found in the
Chinese language
https://www.fluentu.com/blog/japanese/onyomi-and-kunyomi/
The on'yomi (音読み, [oɰ̃jomi], lit. "sound(-based) reading"), the
Sino-Japanese reading, is the modern descendant of the Japanese
approximation of the base Chinese pronunciation of the character at
the time it was introduced. It was often previously referred to as
translation reading, as it was recreated readings of the Chinese
pronunciation but was not the Chinese pronunciation or reading itself,
similar to the English pronunciation of Latin loanwords.
https://en.wikipedia.org/wiki/Kanji#On'yomi_(Sino-Japanese_reading)
Japanese Names
Nanori (名乗り, "to say or give one's own name"; also, by extension
"self-introduction") are kanji character readings (pronunciations)
found almost exclusively in Japanese names.
In the Japanese language, many Japanese names are constructed from
common characters with standard pronunciations. However, names may
also contain characters which only occur as parts of names.
https://en.wikipedia.org/wiki/Nanori
Kawazu is a Japanese surname.
Analysis & Readings of 河津掛け
Having provided some background information on how Japanese uses kanji, I will now provide an analysis of 河津掛け.
The first thing to note is that 掛け(-gake) is used to form the names of a number of moves in judo, referring to a “hook”. Since this part is not in dispute, that leaves us with 河津.
河津 in Chinese
The word “河津”, in Chinese, means “Héjīn” (a county-level city in Yuncheng, Shanxi, China).
The character “河”, on its own means “river” and is pronounced “hé”.
The character “津”, on its own means “ferry crossing” and is pronounced “jīn”.
河津 in Japanese
The fact that the reading of these kanji is “kawazu” means that we are not dealing with an on'yomi reading, which would have to sound something like “Héjīn” and it isn’t even close. The on'yomi reading of the first character is “ka” and there is no on'yomi reading for the second character.
The character “河”, in Chinese, means “river”, while the Japanese word for “river” is “kawa”. So, what we have is a kun'yomi reading here.
The character “津”, in Chinese, means “ferry crossing”, while the Japanese word for “haven, port, harbor, ferry” is “tsu”, which is close to “zu”. So, a kun'yomi reading is a possibility. Another possible reading is a nanori reading, which is for the formation of surnames. The nanori reading of “津” is “su”, “zu”, or “tsu”.
We either have a fully kun'yomi reading, which is to say a Japanese translation of the meaning of “Héjīn” from Chinese. Or we have a partial kun'yomi-nanori reading, where a surname has been formed of the Japanese translation of “Hé” and -zu.
蛙 in Chinese
The word “蛙”, in Chinese, means “frog” and is pronounced “wā”, although in Old Chinese, it was pronounced /*qʷraː/ or /*qʷreː/.
蛙 in Japanese
The character “蛙”, in Chinese, means “frog”, while the Japanese word for “frog” is the more everyday “kaeru” or more poetically, “kawazu”.
The ultimate derivation is unclear, with numerous theories. Some of
the leading ideas include:
Cognate with 帰る (kaeru, “to return (to a
point of origin)”), from the way that some species of frogs return to
their birthplace to spawn
Cognate with 孵る (kaeru, “to hatch (from an
egg)”), in reference to tadpoles
https://en.wiktionary.org/wiki/%E8%9B%99#Japanese
Conclusion
While the character “蛙” is pronounced “kawazu”, this is not the character used in the surname Kawazu or in the name of the term kawazu-gake in Judo. Essentially, what we have is a homophone and the ambiguity can be resolved in writing because “河津” and “蛙” are heterographs.
I would suggest that rather than the name being “derived from the frog (kawazu) because of the shape of the legs”, that instead, in the 12th century Choju giga (Scrolls of the Frolicking Animals), the name of the technique was deliberately misused alongside the image of a rabbit and frog wrestling for humorous effect, since it was effectively a kawazu applying a gake (it could have been any kind of gake, but it happened to be a kosoto-gake), rather than a kawazu specifically applying a kawazu-gake.
