I don't know if there is a way you are supposed to do this. When practicing Scales vertically (like in a position), you know how you assign each finger to a fret? While some of the five positions stay within those four, others do not. When they do not, do you:

a) shift your hand so that your first finger or fourth finger are not hitting that note, or do you
b) stay within position and just stretch?

As I write this, I can kinda see that it might be a subtle difference if you start doing your scales fast enough.

My roommate was playing his ukulele, which he got a couple of months ago. Sometimes he likes to throw fingers across the fretboard or add his finger to known chords and ask me what chord that is. So he added a finger to his F mjr (2010) and made it (2110). Trying to figure out what chord it was, we looked at it as either being a kind of Amjr or a kind of Fmjr. I think, doing it in my head though, that this chord is both an Fmjr-b6 and an Amjr-b6, and they're both missing their 5th! The notes are A, C#, F, A. Sorry for putting this story in the wrong spot, I hope this interested somebody ahaha.
Stay in position and stretch will be the way you see most people do it. Shifting your whole hand position takes too much energy and concentration.
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Which position system are you talking about? The CAGED system?

Say in the "A" shape position, the first one on this image -

You'd play through the position finger per fret then shift your hand up a fret for the notes on the B string. For the D shape, similarly you play through and shift back a fret on the D string.

I really don't understand how you've phrased option a but it's definitely not correct to stay in position and stretch.

Regarding your second question, it's an augmented chord. An augmented chord contains a root, major 3rd and augmented 5th (same sound as minor 6th). Interestingly, an A augmented chord and an F augmented chord contain the same notes (although they would be "spelled" differently).

Also, you don't write "mjr" to indicate a major chord. "A" means "A major".
Thanks Freepower, that makes sense. That seemed to make the most sense to me! Also thank you for that picture, it's more clear than the positions picture I've been using.