I have created a web based application, registered a domain name, and
hosted it on the Internet. Can I say it's a one man startup?
No, it is a dude with a website, not a business so not a startup.
Do you have a busines license? Registered as a business?
If not - it is not a startup. It may be a startup in foundation, but you can not write an invoice as business.
The moment you register as a business - it is a startup.
What will it be if I just host my app on the Internet (not signing any
In most cases: An illegal business. Same like opening a restaurant without proper paperwork.
Any advantages or disadvantages?
Well, except the fines the government may hit you with?
You should get a business license, by whatever means that is done in your jurisdiction, if you ever want to write an invoice.