初めまして vs 始めまして

In general, one of the phrases that a Japanese learner learned in the very first stage is “はじめまして”. However, in terms of Kanji, which one do you think is appropriate for this phrase: 初めまして or 始めまして?
 
Personally, I thought it should be “初めまして”. The reason is that… well… somehow… I don’t know… it should be that way. (embarrassed…) However, there are different explanations in different dictionaries. Some say that it should be “始めまして”. In contrast, some say “初” is the way to describe the “time”, and “始” is to describe “things” or “event”. However, they also say there is no an absolute answer for that.
 
After the investigation, my understanding is that there is a trend for the usage whereas both Kanji are correct for “はじめまして”.
 
When it comes to the meaning of “start”, you should use “始”. In contrast, when it comes to “the first time”, you should use “初”. Let’s practice by the following examples.
 
9時ちょうどになったら、テストをはじめてください。(始めて)
 
はじめてこのセミナーに参加される方は、ここに名前を書いてください。(初めて)
 
先月から日本語の勉強をはじめました。(始め)
 
日本語の勉強は はじめてですか?(初めて)
 
See you next week!