The word in your source is marriage not engagement, (i do not think it is sposificly on the birthday)
The numbers are not sposific (18 just means with the ability to learn the whole Torah)
You can not make practical decisions from the Mishnah, you need to know the reason of the law
The reason of the number 18 is
That a person should first learn and know the whole Torah and then get married, (even though the mitzvah on having children is greater then learning Torah, and he is obligated to do it from age 13, it is permitted to delay it (for him to be able to know the whole Torah ).
If you See the "Brisa" you will see that in those days they started to learn the written Torah at age 5, (and they learnt it for 5 years to know it good).
then at age 10 they learnt the Mishnah (halochos).
then at age 15 they learnt Talmud (the reasons of the halocho)
so by age 20 they knew the whole Torah and should use there time making a living for there family, and learn only part of the day.
The number 18 comes because 2 first years of marriage the yoke of supporting a family is not so heavy yet and you will have time to finish and know the whole Torah by age 20
From Bais shmuel 1.3
It seems that only the Bach holds that 18 years means 18 full years (not counting year 0) the rest say that 18 means after the 17th birthday
The practical halacha is
you should not get Married until
you know the whole entire Torah, or until
you do not have your yezer horo under control (since it is better to learn less but with holiness (without the yeser horos control))