- Powered by nanoStudio?
This means the website in question uses nanoStudio's product to function.
- Design by nanoStudio?
This means "the design of this web page was created by nanoStudio" and is more often used by groups that specialize in design.
- Designed by nanoStudio
This means "nanoStudio designed this website" and is basically synonymous with the previous statement. It's a matter of taste.
- or other expression like present by?
This marks nanoStudio as the owner of the website, which is likely true, but isn't clear yet. If you're building a website for hire, this one is not appropriate.
Personally, I'd lean towards "Design by" if you only did the design and nothing more, and "Designed by" otherwise.
Hope it helps, good luck to you.