there are two forms. one is the short form issued for ID. that's what obama put out. the other is the long form which looks pretty much like anyones BC. you are correct, it is called a certificate of life birth also, but it is much, much different.
there has been word that hawaii has informed all that they will no longer issue the long form on request. they will only issue the short form. i would love to know when that decision was made, but have not found info on it.
i do understand not being able to find a birth certificate. it's hard to keep track of stuff when you don't need it every day. however, when the question was first raised, and there was enough to make people wonder, it should have been laid to rest. it is a requirement for the job. "i can't find it" wouldn't work for any other job.
Back in the day when Social Security was 1st introduced a large number of people couldn't even produce documentation of birth through a then accepted form of record called the Family Bible. Social Security had the policy, and still does I believe, that in lieu of a document of birth the written testimony of 3 relatives older than the person applying for benefits is required. This is then accepted as proof of birth and citizenship.
So a Birth Certificate or Record of Live Birth isn't even necessary to confer citizenship.
It is through the Family Biblical records and testimonies of relatives that my grandparents and great-grandparents qualified for Social Security benefits when they turned 65.