Need a primer / undercoat to suit various wood types

[House7000]

Hoping for some advice. I have recently had a new porch door frame added in with frosted glass side pane, oak door etc. The door is simple enough as I'm using osmo on that but the frame and everything else around the door and glass etc I need to prime. Its been made of a combination of wood types. Think I have some softwood, hardwood and mdf in there so I need a primer / undercoat that will be good for all.
It will be painted white afterwards.

Any recommendations? I've seen zinsser BIN recommended but it seems a bit messy / if it gets on the glass in anyway might be a hassle to get off.

The door is an internal porch door so isn't exposed to the elements if that helps with recommendations.

Cheers

 

[Mr Chibs]

Blue can zinnser, I’ve just used it to paint water based on top of old oil paint, it’s thin-ish to paint with, but has done the job.

You can use wire wool 0000 grade to take marks off your glazing, just go round with the frog tape first.