Different composers have different ideas about notations for accents. Also, the meaning of the accents changes over time. In Türk's Klavierschule of 1789, he writes that notes marked ^ "must be played with somewhat greater strength," while in Beethoven's time period the sforzando marking (generally sf at the time) is considered by contemporary theorists such as Czerny and Hummel to mean "sharply struck." (I don't find the use of ^ in Beethoven's music, which is not to say that it isn't there.)
If you want to dive into the subject a bit more, you can have a look at this article. If you want to take a deeper dive, have a look at this dissertation on Beethoven's use of accents, which, though long and thorough, isn't heavily technical.
At some point, you have to make up your own mind about what the composer means. If you find both of the two notations in Glorfindel's post in a piece of music, then I'd say his rule applies.
In the Satie piece you mention, I'd certainly say that the marking means to put a strong accent on the first beat. The rest of the notes should die away, too, but not starting as loud as the first one. More like from about mp to pp.