I got a non 80s looking perm when I was a teen. I went to an older experienced hair stylist. We looked at pictures, talked it through, and as long as you're not expecting a natural curl pattern you should be fine. The 2nd picture is natural curls and you can't get that from a perm, however your third picture is pretty close to what I got with my perm. My perm was a good experience and people with naturally curly haired actually thought mine was natural because it looked close to the 3rd picture.
Some things to know:
When you have permed hair you can't shower for a few days after getting it done because the perm needs to set. If you go home and shower after getting a perm you will literally be rinsing it out before its able to completely set.
They're are different sized perm rods that can be used. The smaller the rod the tighter the curl is. The bigger the rod the looser the curl is. Since your hair struggles to hold curls as is I would suggest you don't get anything too loose.
After getting my perm my hair stylist made sure I started using a moisturizing shampoo and conditioner. The stuff I used at the time was called moisture maniac, but I don't think they still make that. She said I needed to make sure my hair had lots of moisture to avoid frizz and I never got any by following her instructions.
When you get your perm it will probably be tighter looking than you expect, but it loosens up over time and definitely a little bit after the first month or two. I was told that it's better to have it be a little tighter at first because then it will last/hold its shape longer. My perm was still going strong a year after I got it. It was looser than what it started as, but it was still very pretty. I happen to have hair that holds curls/waves really well which might be why my perm held on so long. I think most perms last between 3 to 6 months depending on the person.
Next, keep in mind that if you wake up one day and decide you don't want your perm anymore you have three options:
Just let it grow out and live with whatever it looks like with your natural hair texture growing in
A chemical treatment at a salon to get your hair straight again and remove whatever is left of the perm
Cut it off
On the other side of the coin, if you love it and you want more curls again once the perm gets looser or starts growing out too much then you can always get another perm as long as you took good care of your perm/hair by using the right products. If you do get a perm consider rinsing your hair with cool water at the end of every shower (its great for fighting frizz and locking in moisture).
If I were you I would make sure to do you're research before getting a perm and sit on the idea for a month or two to make sure you really want it and that's it not something you're doing on a whim. I would Google perms, look through pictures, and see if they're are images of perms that appeal to you. Also consider googling "digital perm" as well (a digital perm is a more modern take on old school perming). I think by googling "perm" as well as "digital perm" it can help you narrow down which is right for you and help you better know your options.