if a charachter wants to modify a leather coat to be like a lined coat, would this lined leather coat get an armor rating of 4/3 for synthetic leather or 4/4 for real leather?
would you say that it makes any difference?

I'd say the distinction is irrelevant. Just buy the regular lined coat or secure long coat and have it look like leather. (actually cheaper than real leather, I notice)

Seems like a lot of work for one measly point of impact armor.

Style over substance, my friend, style over substance.

Treat it like a normal lined coat. The fact that it's leather is for roleplaying, not numbers.

Ask the GM. Some allow bonus for listed armor made out of leather.

But would a leather jacket, even one made out real leather, put up such a massive resistance to flechette bullets?

I was just thinking about this the other day. I decided not to bother the GM for an Impact point and bought an armored jacket that I describe IC as "snyth-leather"

The leather on all the critters I have ever shot never seemed to help them any.

I'm sure a Jacket (0/0) made out of Troll skin would prevent Flechette ammo from staging up.

Would it? Isn't the natural dermal plating beneath the layers of skin that would be used to make a coat?

I don't see why a real leather armor jacket/coat wouldn't have a higher impact rating than a regualr one. Look at the regular jackets made of leather and synth-leather for instance. Synth-leather has an impact rating of 1 while real leather has a rating of 2.
I'd just reccomend adding the price of the two together. Thus if you want an Lined coat made of real leather it would cost you 1450¥ (lined coat 700¥ + 750¥ real leather jacket) and have a rating of 4/4.

If you want the two, armor from Real Leather and Lined Coat, using the armor stacking rules as normal. Wearing just that, it'd give you a total armor rating of 4/3 (4 Ballistic, 2 + 2/2 = 3 Impact). Won't get you into any trouble unless you have a QUI of 3 or less, is highly Concealable. Mostly useless if you wear additional armor.

Considering how massive a heavy leather coat with soft body armor inside it is, I think considering it stacked armor makes sense. And it allows a character to get the Real Leather and Lined Coat armors, which might otherwise be really difficult. Actually making it as good a form of protection as an Armored Vest with Plates (ie 4/3, but not considered stacked armor) sounds a bit silly to me.