What is this SMART page?
He really was sparse there. What he meant was the SMART report of the hard disk.
Some information saved on the HD by SMART, that I can access with a SMART tool like smartmontools?
SMART do not save anything in your drive, but more like S. M. A. R. T. (Self-Monitoring, Analysis and Reporting Technology) is a set of test that your drive do regularly to prevent failures and improve performance. You can access this data with smartmontools as you have said, since this information is independent to your OS. It shows all sorts of data about relocated sectors (as he mentions), spins ups/downs times of the drives, etc, etc.
A page (tab) in his SpinRite software?
Yes. This was what he was referring to. SMART page, SMART report, SMART visualized, SMART whatever. When you talk about S.M.A.R.T. you are talking about the Self-Monitoring, Analysis and Reporting Technology included in all (modern) HD's. When you add another substantive is most likely to refer to a tool that reads/manipulate this data.
In any case, can I see, in any way, what sectors are marked as bad, without using SpinRite?
Yes. You can. The format the data is presented depends of the tool. Also, some vendors add their own CODES, so, sometimes you need their tools to read their codes. But! You don't need their tools specifically to read the SMART data of your Hard Drive.
Some insights about SpinRite
SpinRite seems to interpret the RAW data differently to your "Average Tool", taking in consideration other sets of parameters which they consider important, relevant, informative or preventive. Their tool just postprocessing the data is already there and shows it into an easy to understand way. You may have an idea of what could go wrong if you read the RAW data, SpinRite process the data and tells you exactly what (they consider) is bad.