“Saisen” is the money that you offer on a normal jinja visit, dropping it into the offering box before you pay your respects. There is a common custom of offering five yen (about 5c US), because the Japanese for “five yen” sounds like the Japanese for “good connection”, but jinja sometimes say they would like you to offer at least ¥100 (about $1). In any case, the amounts are much smaller than those associated with a formal prayer, which start at about ¥5,000, and are called “hatushoryō” or “tamagushiryō”.
There is a debate over whether you should throw these offerings into the offering box, or drop them in gently. Sometimes, throwing the money is the only way to get the coins into the box, but normally you could drop them in gently. The May 3rd issue of Jinja Shinpō reported a talk given by Professor Shintani, an ethnologist at Kokugakuin University, and formerly at the National Museum of Japanese History, on the meaning and history of both saisen and hatsuhoryō.
He emphasised that they were different, and that saisen had developed from “sanmai”, where rice was scattered as an offering. He said that the original meaning was twofold: an offering to the spirits of the land and to hostile spirits, and purifying and driving out hostile or evil spirits through the power in the rice. When this developed into saisen, it kept, he said, both meanings, so that it has the meaning of purifying the person scattering the money, as well as being an offering.
That history rather suggests that, of the options, throwing the money is “more correct”. Hatsuhoryō should be offered reverently and calmly, because it is entirely an offering to the kami, presented as part of the matsuri. Saisen should be scattered, which in this case means thrown into the offering box. Still, customs change, so if a particular jinja asks you to put the money in gently, that’s what you should do there.