No big difference and also no official rules.
Nihon : formal/official cases a lot (not for all)
Nippon : other cases widely (even using in formal/official cases)
Word origin of "Nippon" older than "Nihon" fr over 1000 years before already,
but no official records about how to change to Nihon and ways of spread.
So even Japanese who born in Japan can not read properly
if written in Kanji/漢字 "日本" except common/major words and names.
日本 : Nihon/Nippon : Japan
日本国 : Nihon-koku/Nippon-koku : Japan as nation (but even reading of official country name is not fixed)
日本語 : Nihon-go : Japanese as lang (no use Nippon-go)
日本人 : Nihon-jin : Japanese as race (also using Nippon-jin case a lot)
日本の : Nihon-no/Nippon-no : xxx of Japanese, Japanese xxx, xxx in Japan and more.
日本史 : Nihon-shi : History of Japan (no use Nippon-shi)
日本製 : Nihon-sei : Made in Japan (Nippon-sei is not wrong, just using not a lot)
Most names are fixed,
日本海 : Nihon Kai (Japan SEA as nautical chart, Sea of Japan)
JR東日本 : JR Higashi-Nihon(JR-EAST, major railway company in east)
JR西日本 : JR Nishi-Nihon(JR-WEST, major railway company in west)
近畿日本鉄道 : Kinki Nippon Tetsudo (one of major Private railway company in west)
西日本鉄道 : NishiNippon Tetsudo (one of major Private railway company in Kyushu)
日本航空 : Nihon Koukuu (Japan Airlines Co., LTD. =JAL）
日本通運 : Nippon Tsuu-un (NIPPON EXPRESS CO., LTD.)
日本科学未来館 : Nippon Kagaku Mirai-kan(National Museum of Emerging Science and Innovation)
日本近代文学館 : Nihon Kindai Bungaku-kan(The Museum of Modern Japanese Literature)
ニホンザル : Nihon Zaru (Japanese macaque : Macaca fuscata, monkey)
ニホンカモシカ Nihon Kamoshika (Japanese serow : Capricornis crispus)
朱鷺 : Toki (Crested ibis, binomial name : "Nipponia nippon", rare bird)
Some major companies name using both.
日本テレビ : JP : Nihon Televi(sion), but EN : Nippon Television
日本生命 : JP : Nihon Seimei, but EN : Nippon Life Insurance Company(Nissay)
日本ハム : JP : Nihon Hamu(Ham), but EN : Nipponham
These are one of rare cace.
日本橋(bridge) in Tokyo is Nihon-bashi, but 日本橋 in Osaka is Nippon-bashi.
(map as town/mail address, rail stn, bus stops using special Roman "Nihombashi", "Nippombashi")http://mamechishiki.aquaorbis.net/wp-content/uploads/2015/07/nihombash...http://livedoor.blogimg.jp/kansai_railway/imgs/a/b/abaeab73.jpg
日本銀行 (Nihon/Nippon Ginko) : Bank of Japan, but only in banknotes(bill) using "Nippon".https://www.boj.or.jp/en/note_tfjgs/note/valid/issue.htm/
(RED oblique lines and 見本 : sample is copy protection)
日本女子大学 : Nihon Joshi Daigaku (Japan Women's University）,
but school nick is "PonJo" Nippon based, it is also very famous.
So, No one can explain it perfectly.
"Ippon!(1本, 一本)" is NO related as sound of pronunciation.
That 本(hon/pon/bon) is one of "counter word(助数詞)".https://en.wikipedia.org/wiki/Japanese_counter_wordhttp://www.punipunijapan.com/counting-in-japanese-hon/
(6, 8, 10,,, also 100本 : hyappon)
If yr hypothesis is right, we also have to read so in other numbers cases all.