I don't understand this! Isn't TAX residency always for a full calender year. Even if you only get to Cyprus 2 Month before the year ends, you should be considered TAX Residence for the full year.
Even if they can claim you are still TAX resident for the year, they can't say therefore your newly created company is German and now you have to pay an exit Tax, right?
Well, you are mixing up a couple different aspects, let's untangle it.
In Germany the tax year equals the calendar year. So if you leave during the year then you still have to file a tax declaration for that year and declare all worldwide income. However, income that you received after ceasing to be a tax resident will only increase your marginal tax rate but not be taxed itself.
This is a bit complicated, so here a rough example.
Say your income for 2020 is 5.000€ / month and you left Germany July 1st. You will have to file taxes for 2020 and you will have to declare 60.000€ in income. The 30.000€ that you earned in the second half of the year will not be taxed in Germany, however the percentage that you pay in tax on the 30.000€ from the first half of the year is calculated based on your total salary of 60.000€.
Basically, normally for 30.000€ income you would pay about 14% income tax, but because your total income is considered you now pay 25% of the 30.000€, so 7.500€.
Concerning getting a second tax residency.
If you move to another country where you would also become tax resident in 2020 (e.g. Cyprus from your example), then you would for 2020 actually be tax resident in two countries and would need to consult the double taxation treaty to see how this situation is handled and who gets to tax your income.
Concerning the exit tax.
Exit taxation for entrepreneurs provides a good overview of the German exit tax and if it applies to your situation.
To your specific question - no, when you leave Germany (even during the tax year) and then set up a new company afterwards then this new company will generally not be covered by the exit tax.
(I am just a random guy on the internet, not a tax advisor. These are rough examples. Please discuss your situation with a tax consultant to make sure that you are fully compliant.)