Generally with arpeggio's proper the sequence is played in order, that is typically how you hear them used. Albeit starting on any degree and changing direction at will. Generally what is happening in jazz and blues as a part of the lead line or melodic thing, more scalar thinking.
If you hold a chord and just kind of pick out the notes of that chord in it is often called "arpeggiating" the chord. That is more chord grip shape based, more rhythm thinking, instead of strumming you pick out notes, more harmony thinking. Generally what is happening in folk finger style.
There is a lot of overlap though, the two concepts melt together around the edges...