i'm no expert, but the fact that you're asking about a "birch" wood snare, makes a huge diff, no?

not as resonant, so you'd need an extra half inch, eh?

just a thought.

my vote would be a 14x4 or better yet a 14x5.  tunes higher, for more crack.

I like 6.5's because you can always tune them up if you want a higher pitch.  There always seems to be added depth to a 6.5.  I only use shallower drums where I need more sensitivity.  The birch drums usually don't h ave the sustain the maple ones do anyways but it should be plenty sharp with plenty of crack any depth you get.  Yamaha makes a ton of different snares, so you would be buying this to complete your kit.  But you just might want to buy say, an Anton Fig or Steve Gadd model, or whatever really turns your ear on.

If I was doing really soft stuff I would go 5.5 or below...everything else 6.0-6.5.  It seems if you go too deep in wood the drums seem to really "dry out".  I do not like piccolos for loud stuff, but I like the john bonham, alex vanhalen snare sound and variations there of.  You can puke now or love it...I'll take the latter.  I love the tone.

13" in diameter drums are not my bag...they never sound "right" to me, but I am a traditionalist.   Too "boopy" is the only way I can explain it.  I guess I like the spread in tone a 14" gives me.  If I have to focus it more, I put some tape on it and forgeddaboutit.

AS always YMmV