Technically speaking, if you strum the Am chord from the open E, you're playing a Am/E chord (so need because it's an A minor chord with an E note instead of the root note as its lowest note). However, as the notes that make up the Am chord are A, C, and E, you're still playing an Am chord. The only reason all six strings are being played at the level of skill in that particular lesson is to keep things simpler in order to concentrate on strumming.

When you get further into playing, and much much further into understanding the theory behind chords (if that's part of where you want to go) you will start to see that chords don't necessarily start on the root note, and may be ignored completely if you have a bass player taking care of them.

There's a lot to learn, but you're definitely in a good place to do it.