Florida unmarried ladies, is this true at all?

My FUTURE mil (whom I adore and she adores me), informed me that in other states(she has a second house in South Carolina, she spends summers there) if u are unmarried, even if u are in love, on good terms and living with ur SO, if u are not legally married, on the birth certificate of the baby well say " baby out of wedlock" . Now! I didn't want to belive it, but when her niece(19 years old) texted her own birth certificate to her and she showed it to me, I was floored. It's in fact true, it said across the whole thing that the baby it's out of wedlock.

I was in complete shock. That's SHAMING UNMARRIED WOMEN! How is this legal?

Is this also true in FLORIDA? !

My SO and I got engaged a month before finding out I'm pregnant.. and once we found out, we decided to wait til after the baby since there's so much stress in our lives right now. And now, I'm due in October, and I really do NOT want to go to court house and get legally married in a hurry, with no plans whatsoever, just so it doesn't say that my baby was born out of wedlock on her BIRTH CERTIFICATE...

Do any of u know if this is true in Florida?