Highlighted it for you.
You're lucky I'm bored.
It works like this (in theory - in practice they probably don't care in Cyprus):
1. Is the company managed and controlled in Cyprus (no matter where it was registered)? If yes, the company is tax resident in Cyprus.
2. Is the company incorporated in Cyprus and not tax resident in any other country (under that country's rules)? If yes, the company is tax resident in Cyprus.
3. If the answer to both 1. and 2. is "no" (i.e. the company is not managed and controlled from Cyprus and, if it's a CY company, some other country claims it as tax resident under that country's rules): The company is not tax resident in Cyprus.
4. If the company is tax resident in Cyprus (either 1 or 2 answered with "yes"), then all income that the company has is taxed in Cyprus - unless a tax treaty says otherwise.
5. If the company is not tax resident in Cyprus (both 1 and 2 answered with "no"), then only the income that is earned through a PE in Cyprus is taxed in Cyprus.
What does this mean (in theory - the practice can be different in Cyprus):
Company A is registered in the
Seychelles.
But its owner and director lives in Cyprus all year. He rents an office in Cyprus in the company's name where he meets clients and does all the work for the company.
He also has an employee in Japan (there is
no tax treaty with Japan). The person is employed directly by the
Seychelles company and there is an office in Japan that is rented in the company's name where the employee meets clients and works.
This company is clearly managed and controlled in Cyprus and it will be taxed just as if the company was registered in Cyprus.
All company profits, including the profits generated by the employee in Japan, are taxed in Cyprus. On top of that, the profits generated in Japan are probably also taxed in Japan under the Japanese rules (there would likely be double taxation), but that's not our focus here.
Company B is registered in Japan.
Its owner lives in Japan all year. He rents an office in Japan in the company's name where he meets clients and does all the work for the company.
He also has an employee in Cyprus (there is no tax treaty with Cyprus). The person is employed directly by the Japanese company and there is an office in Cyprus that is rented in the company's name where the employee meets clients and works.
This company is clearly not registered in Cyprus and also not managed and controlled from Cyprus - the employee in Cyprus is just an employee, he doesn't manage the company.
There is a permanent establishment (office) in Cyprus. For this reason, Cyprus can tax all profits that the "CY office" makes for the company. However, since the company isn't tax resident in Cyprus (since it's not managed or controlled from Cyprus), the income generated in Japan or any other country is not taxed in Cyprus.
Japan might still also tax the company profits generated in Cyprus (not sure about the Japanese rules), then there could be double taxation. But again, not our focus here.
I hope it's clearer now.