Piers,
As you found, it looks like a takohiki.
Takohiki (蛸引) type is mainly used in Kanto area and yanagiba (柳刃) type is used in Kansai area. Both types are used to cut sashimi.
There were many samurai dramas after the early 1900s. Most of them were entertainment works with questionable historical accuracy. As one of nonsense settings, samurai use their kozuka as trowing knives in films. Ordinally people still believe such acts even nowadays.
She might be merely one of them. And I think that she confused kogai, kozuka/kogatana and kankyuto/bashin.
Samurai TV drama in 1972, false use of a kozuka.
FYI, comparison of swords without a hi and with a hi.
Watch from 5:30.
日本刀に彫ってあるミゾ(樋)はどんな意味があるの?溝がある刀とない刀の違い(What does the groove on a Japanese sword mean?)
The seller’s description seems to be correct.
脇指 無銘 (兼房) - Wakizashi, unsigned ((Thought to be) Kanefusa)
(切付銘) - Kiritsukemei
寛永十九年三月ノ初五 – Kanei 19th year (=1642), the 5th day of the 3rd month
於武州江城□脇毛深守土中五寸 – At Edo in Bushu, cut through shoulders and into the mound by 5-sun (=15 cm).
石原小□彦成正(花押) - Ishihara ??? Narimasa/Shigemasa (+ kao)
長一尺二寸九分強 – Length: A little more than 1-shaku, 2-sun, 9-bu
守護短刀 壹振 – Gurdian tanto, one
日本砂鐵鋼業株式會社以精錬鋼長七寸謹作大坂住月山貞勝 – Using the steel refined by the Japan Iron Sand Steel Industry Co., length 7-sun, respectfully made, Gassan Sadakatsu in Osaka.
The note on the paper seems to be poems. I think that they are not directly related to the tanto.