Rosh Hashana 10a - When is the Bar Mitzvah?

Tosafos (d.h. ben) says that sometimes we don't use yom echad b'shana chashuv shana, but we require full days mei'ais le'ais. The example Tosafos uses is becoming Bar and Bas mitzvah, where we say 13/12 and 1 day - "that one days is to complete the mei'ais le'ais". I understood Tosafos to mean that really you need 13 complete years to the hour, but since it is difficult to remember the time you were born, we require 13 years plus a day. So, even if one was born adar 7th toward evening (right before shekiah of adar 8), by requiring him to wait the extra day (bar mitzvah would be ohr l'adar 8), we would insure that we have 13 complete years. But, if someone does remember when they were born, they would not have to wait the entire extra day (unless we are to assume that chazal made a lo plug). Obviously this is not the predominant custom. I found that the magen avraham 53:13 quotes from the Bach that as soon as your 13th birthday comes, you are bar mitzvah even if you were born later that day (the extra day is just to the exclusion of using miktzas hayom kikulo on the last day before your birthday). The Shach in C.M. 35:1 (mentioned in Magen Avrohom) cites the divrei chamudos who understood Tosafos in R.H. like I explained, and the shach argues that when tosafos says mei'ais le'ais they don't really mean down to the hour, he therefore concurs with the Bach. I also found that the Mishneh l'melech (Ishus 2:21) explained the Tosafos the way I did, and that according to this Tosafos the Bar Mitzvah would be on your birthday at the hour of the day you were born. But, he cites many opinions who hold like the Bach (Tosafos in Nida and Erchin, Rashbah and mashmaos of gemara in Yevamos), and therefore rules like the Bach (not like our Tosafos).