Is it halachically the case that on a 2 day Rosh Chodesh, the upcoming
month is considered as if it has begun even on the first day?
The basic answer is yes, unless specific language is given to differentiate. Thus if one says the start of Adar or Rosh Chodesh Adar, the the first day of Rosh Chodesh is meant. If one explicitly says the first of Adar then the second day of Rosh Chodesh is meant.
As an example, the Rambam says that if someone makes a neder for "this month", the thirtieth day of the month (first day of Rosh Chodesh) is treated as the next month and the neder has expired. Thus we see that when someone speaks of "Rosh Chodesh" for halachic purposes, then we use the first day of Rosh Chodesh as beginning the month based on the principle that the Torah speaks Belashon Adam under normal circumstances
Rambam Nedarim - Chapter 10
שאני שותה חדש זה אסור בשאר ימי החדש אבל ביום ראש חודש יהיה מותר אף על
פי שהיה חדש חסר שאני טועם חדש אחד אסור שלשים יום גמורים מעת לעת נדר
חדש סתם אסור שלשים יום מעת לעת מספק:
[When one takes a vow, saying:] "I will not drink [wine] during this
month," he is forbidden in the remaining days of the month. He is,
however, permitted on the day of the following Rosh Chodesh even if
the month is lacking.8 [If he took a vow, saying]: "I will not drink
[wine] for an entire month," he is forbidden for 30 full days. [If he
said]: I will not drink [wine] for a month," he is forbidden for 30
full days because of the unresolved question.9
A month which is lacking is a month of 29 days [as opposed to a month
of 30 days; see Hilchot Kiddush HaChodesh, chs. 1-3, which discusses
the principles determining when a month is given only 29 days and when
it is given 30].
The commentaries question why the Rambam (based on Nedarim 60b) speaks
of the month being lacking. Seemingly, it is quite obvious that if
there are only 29 days in a month, one would be permitted on Rosh
Chodesh in the next month. The new month has already begun. A point
that has to be made is that even if there are 30 days in a month, one
is permitted to partake of wine on the thirtieth day. Since it is Rosh
Chodesh of the coming month, the vow has concluded even though the
date is the thirtieth of the previous month.
The Radbaz explains that this in fact is the Rambam's intent, even
though his wording is somewhat difficult to explain in that manner.
This interpretation is reflected in the wording of the Shulchan Aruch
(Yoreh De'ah 220:4). The Or Sameach offers a different interpretation,
stating that when a month has only 29 days, sometimes the conjunction
of the sun and the moon does not take place until the first day of the
new month. Even so, since it is already Rosh Chodesh, the vow is