Hello so In my first playthough I went and bought the mark 2 synth helmet, and I was surprised of how cool it looked because it was closed like the field helmet but battle worn. And in my second playthough I go and buy it again but for my surprise the helmet lid is open now .

Do I have to belong to that faction to have it with the lid closed??

There's multiple versions of the helmet. You may have gotten the 'synth field helmet' the first time and that is full faced

no its the mark 2 synth helmet, its full faced and looks battle worned

There's 3 levels of armour depending on your character level. For Raider, Leather, Metal, Synth and Combat armours there is standard which adds no prefix to the name, then there's Sturdy (Sturdy Raider, Sturdy Leather, Sturdy Metal, Sturdy Synth, Sturdy Combat) which is a more visually complete set and last there's Heavy which is more visually complete still. So the open faced Synth helmet will be the standard piece, closed and worn is sturdy and the closed and shiny will be heavy.

My first play through I only started seeing Sturdy in the high 30's, early 40's. I'm guessing the Heavy variant doesn't start appearing until after you get to level 50.

I was level 39 when I went to the institute shop, its a legendary helmet, and in my first playthough I was level 27-28 cant remember exactly