The Gemara (Rosh Hashana 2b) tells us that there is logic to say that only the last day of the year is considered a whole year but not the first day. The discussion there is about a king who started his reign on the last day of Adar. In such a case one can argue that when the New Year for kings arrives on the first day of Nisan, this king has already completed one year of kings, because he reached the end of the year and he is still the king. But if he became the king on the first day of Nisan one cannot say he has completed a year as a king because the year for kings has only just started.

Kayitz Bari,

Dovid Bloom

Rashi (Rosh Hashanah 2b, DH B'Sof Shanah) is discussing a different scenario. This is a king, regardless of his own personal age, who became king on 29 Adar, which is the last day of the "Year for Kings." This "Year for Kings" is a general protocol for all different kings. If a king had already been king for 34 years, then when the first day of Nisan of the next year arrives he is now considered as being in his 35th year as king. Likewise, the king who started his reign on 29 Adar is considered as being in his second year as king when the second day of his reign, namely the first of Nisan, arrives.

This makes a difference when any citizen in the kingdom prepares an official document. One has to write the year of the king at the top of the document. When the first of Nisan arrives one writes a new figure for the year of the king. If, up to now, we had been writing that it is the 34th year of the king, we now start writing that it is his 35th year. And for this brand new king, who only started ruling on 29 Adar, when we get to the second day of his reign we start writing that it is the second year of his reign.

Kol Tuv,

Dovid Bloom