Can the pinyin for 边 be either "biān" or "bian" in general? It looks like authors choose "biān" or "bian" as the pinyin for 边 haphazardly at dict.cn:


  东边 = dōng biān
  上边 = shàng biān
  右边 = yòu bian
  左边 = zuǒ biān
  旁边 = páng biān
  外边 = wài bian
  一边 = yī biān
  两边 = liǎng biān
  那边 = nà biān
  这边 = zhè biān
  里边 = lǐ biān
  天边 = tiān biān


(There's also 海边, 身边, 内边, and 谁边.)  It likewise seems haphazard at xiaoma.info.

Question: Can the pinyin for 边 be either "biān" or "bian" in general?

This question basically generalizes How come the first tone in the character of 边 in the word 东边 is not written?.  Maybe there's cases where "biān" xor "bian" is the only strictly correct pinyin (or pronunciation).
As I answered in another question you are referring to, those 边s should be pronounced as the first tone biān in standard mandarin except when 边 connotes 'positions' and 'directions', such as 里边 前边 东边 左边 这边 and 那边 in your list. Other than that, the rest are all first tone biān. 

However, in spoken or daily usage, lots of people often pronounced it as neutral tone even for those that should be pronounced as first tone as dictionaries indicate. 

The best way to get it correct all the time is to look up any authoritative dictionaries, such as 新华字典, 现代汉语词典 and etc. Some on-line dictionaries might not be right all the time because they are not well proofread. Usually, they just copy here and there. 

Updates:

After I looked up in the latest dictionary 现代汉语词典， I have some updates on this answer above accordingly.