二月 HLL(me) LHH(NHK)
二月四日 HLL(me) ≧ LHH(me)
四月 HLL(me) LHH(NHK)
四月四日 HLL(me) ≧ LHH(me)
I'm a native and I live in Yokohama since about 50 years ago.
I noticed two things:
The names of the two months concerned have only one syllable. Confusion may likely occur between them when spoken with a small voice or low pitch at the biginning like LHH. Therefore, we seem to become to speak them naturally with loud voice or with high pitch at the beginning of them like in HLL even both pronunciations are allowed.
The pitch is different when saying only the name of the month and when saying the month and the date together.
Even in the latter case, I think that the previous noticing is involved. For example, when 二月四日 is pronounced, 二月 is usually known in the conversation, so what you only have to listen to clearly is 四日. Therefore, 二月 may be allowed to be pronounced lightly. So if two pronunciations are allowed, 二月 may be pronounced in LHH pitches, which is sometimes hard to listen to.
By the way, although 四月 is claimed to be pronounced as LHH in both dictionaries, I think that this is a phonologist's preferred expression, and I usually pronounce the word expressed by this mark flatly without placing an accent anywhere with my intention like LLL not like HHH.
Bonus
In my answer I wrote that in order to clearly convey one-syllable words, we devise to pronounce them in a loud voice or we put a pitch accent on the starting point of the word like HLL, even it is claimed to be pronouced in LHH or flatly.
I think that this device is applied to the Japanese who speak with standard pronunciation and accent including myself, but I would like to introduce different measures that the Kansai dialect takes so that one-syllable words could be heard more clearly.
When the Kansai people pronounce a one-syllable word like "[木]{ki} tree", "[酢]{su} vinegar" or "[手]{te} hand", instead of placing an high pitch accent at the begininng of it, they would enlong the vowel and add the same vowel after the enlonged one like 酢 suh-u.
The way to pronounce it is to lower the pitch gradually from the begining of the word to the enlonged vowel and to raise the pitch suddenly at the lastly added vowel.
In this way, they pronouce a one-syllable word as much as two to three times long.
By doing this, enough time could be secured for listeners to hear the pronounced one-syllable word. I think the Kansai people are clever.
I do not know why, but regarding the pronunciation of 二月 and 四月, I think that the Kansai dialect does not use its good device.
尾高型:1、2、4、6、7、8、10、11、12月 頭高型:3、5、9月 ところが、本来尾高型の2月と4月を、頭高型で[ニ\カ゜ツ][シ\カ゜ツ]と発音する人が増えています。間違えないように「頭高はゴー、サンキュー」と覚えておきましょう。