Is Satori like a stage of stream entry? Is Satori only an intellectual knowing or something deeper than that?
Satori is considered a "first step" or embarkation toward Buddhahood
This is very much what Stream Entry means in Theravada. Everything else in that entry makes Satori appear to be the same thing as Stream Entry.
The meaning of Satori as "seeing into one's true nature" is also the same as the meaning of Stream entry when you consider that "one's true nature" in the context of Chan Buddhism is Nirvana/Nibbana.
Therefore it seems that these two terms do refer to the same thing.
In the context of Zen training, I think the word you are looking for is kensho. The connotations associated with that word point to a fundamental change in consciousness and self perpection that is much more akin the Theravadan notion of stream entry. Satori simply means insight. While is can be used almost synonymously with kensho, it is much more often used to described fleeting, incomplete, and momentary insight.
The dictionary definition of 'Satori' reads:
- sudden enlightenment.
At what stage a Bodhisattva has "sudden enlightenment" is hotly debated in Mahayana Buddhism, and some will weight that stage the same as stream entry, whereas others won't. The point really, in the so called sudden schools, is that with sudden enlightenment the practitioner has realized they are a Buddha, in a key way. Even when there is further to go. This can be "intellectual", though the term 'Satori' is common in zen, which famously discredits all intellectual insight.
I believe that kensho is talked down by Dogen; cf Rinzai Buddhism. Part of that will be secrarian, and partly it will reflect different conceptions of practice. But all zen Buddhism highly prizes "sudden enlightenment".
- If you want to know what stream entry is and how it is achieved then I'd advise reading the Abhidhamma.
- If you want to know what 'Satori' is in zen then practice zen, read its poetry, etc.
- If you want to what 'sudden enlightenment' is in general, read important Mahayana Sastras and commentaries, etc..
I think you'll probably find that they are not treated as synonyms anywhere. Simply because surely all contemporary zen, even, acknowledges that Theravada Buddhism is a different thing.
If you're asking whether a stream winner knows anything someone with Satori doesn't -- or vice versa: give up now. A joke, but anyway Satori is an insight into Buddha-nature, and stream winners have right view in and for Theravada. Someone here can probably detail the qualities of each.