You may need to address this issue in Ca. courts since the birth certificate was issued from Ca.. It's always difficult to amend a birth certificate -- lots of hoops to jump through.
The information is for general information purposes only. Nothing stated above should be taken as legal advice for any individual case or situation.
First, don't worry about it. It is perfectly legal to use any name you like, and to spell it any way you like, except to commit fraud. However, it makes sense to have a legal document stating that your name is spelled the way you spell it. Rather then trying to change your birth certificate, which can be a challenge, I suggest you go to the Thurston County District Court and obtain an order changing your name from Lynn, as indicated on your birth certificate, to Lyn. It is simple to do yourself and inexpensive.
I have attached a link to the court's website about name changes.