I tend to use Google to to research and go through multiple website but do exclusively use AI to solve tech issues. So it really depends on the activity for me.
If that's true, they are using a separate search API to get search results and feed it into a regular Claude API call. The difference here is that Anthropic is integrating it directly, like OpenAI and Google have. It doesn't look like it's in the API yet, but presumably that's coming. Then, as with gpt-4o and the Gemini models, you can make a single API call and it will do the searching for you and incorporate the results.
Can I query multiple vectorized websites at once? Can I export vectorized websites and host them myself? Any chance to export them to a no-code format, like PDF?
You can group as many websites as you want into a collection. Then query that collection.
Not sure what you mean by exporting; you would like to export the vectors themselves? Or just the chunks of text from the websites?
Use the AI to coach you, instead of relying on it as a crutch.