Western nations aren’t funding people to work in Russia.

They are. VoA and Red Cross do function there. Alongside with a thousand of commercial organizations like Manpower and Unilever

Russia requires people who live there and accept funding from overseas to register as foreign agents

Yes and no. Currently, the government can pronounce anyone a foreign agent stating that they are under foreign influence. Money is not the source of this status anymore.

Foreign research initiatives in Russia used to work a little bit different to what you have probably used to. There were, effectively two ways:

1. Foreign organizations grant Russian scientists money for research, but the money is transferred to the Russian government, and the government gives it to the scientists in a year.

2. A foreign organization and a Russian both give a grant to a cross-national group. The foreign one transfers money to the foreign part of the group and the Russian one transfers to the Russian.

Both these ways seem to work right now, so the claim in the article looks suspicious to me.