The vet will be basing the vacc schedule on date of last vacc so s/he probably isn't wrong, and 14weeks is still 3 months.
If you want her to interact with other dogs earlier, consider enrolling in a puppy kindergarten - they are done in secure, sterile environments so the pups don't pick up things.
Also keep in mind puppy socialisation is not the same as 'socialising'. It's about exposure to different environments and objects, not just people and dogs, and there's still a lot of this you can do. Dr Sophia Yin has produced this puppy socialisation checklist
https://drsophiayin.com/app/uploads/2015/12/Socialization_Checklist.pdf . You can do a lot of this at home, introduce her to umbrellas, hats, different types of surfaces, play noises on your computer.
You should also take this time to practice leash walking at home, much easier to get her used to that before you head out into the scary world.
Something I found useful during this stage was to take my dogs to different environments - Tully was tiny so fit in a handbag, Deska I had to carry. We're lucky to have a breed that's easy to carry. Two good places to start are outside schools - so she gets used to the sound of children, and outside small shopping strips (always start small). Not only are they great environments for movement and activity and seeing all sorts of people, but people and children will probably come up to look at the pup, and pat the pup.
Socialisation doesn't need four on the floor, you can start with socialising for people, noises and environments now.
