I am aware that they are not synonymous, but the proposed restriction was too strict and was inconsistent with our entry for はじめ, for example. 

Rejecting this branch so I can approve Jim's amendment.

So basically, you are saying that 始めに and 初めに should be 
treated as both homonyms and synonyms. I'm not saying you're 
wrong, but that's not true according to the references I 
cited and so I would ask that you research this further. I 
wonder if those 英字郎 examples you cite are correct 
illustrations of how these are to be properly used. Examples 
of how they are used in sentences are not necessarily 
examples of how they should be used correctly. OTOH, the 
references I cited give rules for proper usage without 
sample sentences. I just don't like things to be unclear 
especially when there do appear to be rules as to when to 
use one over the other. The best I can tell, the right one 
to chose may have to do with how much emphasis you want to 
place on point in time or order of operations. I think 初めに 
has to do more with point in time and 始めに with order of 
operations, and that is why I split the senses the way I 
did. Can you find actual rules that say differently?

i'm hesitant to call sense 2 a noun.  i don't think the sense restriction is appropriate, as for example eijiro has examples of 初めに being used for the proposed sense 1

@@ -16 +15,0 @@
-<stagk>始めに</stagk>
@@ -18 +17,2 @@
-<gloss>to begin with (used as an introduction)</gloss>
+<gloss>first</gloss>
+<gloss>firstly</gloss>
@@ -19,0 +20,3 @@
+<gloss>to begin with</gloss>
+<gloss>in the beginning</gloss>
+<gloss>at the beginning</gloss>
@@ -22 +24,0 @@
-<stagk>初めに</stagk>
@@ -24,8 +26,2 @@
-<s_inf>in reference to point in time</s_inf>
-<gloss>at the beginning (of the month, year, etc.)</gloss>
-<gloss>in the beginning</gloss>
-</sense>
-<sense>
-<stagk>始めに</stagk>
-<pos>&n;</pos>
-<s_inf>often kana only</s_inf>
+<s_inf>as section of heading of a book, etc.</s_inf>
+<gloss>Introduction</gloss>

"NTC's New Japanese English Character Dictionary" (1993) 
usage notes, and notes in "A New Dictionary of Kanji Usage" 
(1982)

Took out my gikun suggestion as did not appear to be well 
received. Made sense 1 a restricted sense for 始めに. Added a 
restricted sense for 初めに and moved "in the beginning" to 
that. I've seen my proposed third sense (Preface) used with 
both 始めに and 初めに (usually 始めに or kana only), but have 
a feeling based upon "NTC's New Japanese English Character 
Dictionary" (1993) usage notes, and notes in "A New 
Dictionary of Kanji Usage" (1982) that 始めに is technically 
the correct usage and use also matches with sense one.

@@ -11,4 +10,0 @@
-<k_ele>
-<keb>元始に</keb>
-<ke_inf>&iK;</ke_inf>
-</k_ele>
@@ -19,0 +16 @@
+<stagk>始めに</stagk>
@@ -21 +17,0 @@
-<s_inf>元始 is found with gikun reading はじめ in Meiji version of The Bible</s_inf>
@@ -23,0 +20,6 @@
+</sense>
+<sense>
+<stagk>初めに</stagk>
+<pos>&exp;</pos>
+<s_inf>in reference to point in time</s_inf>
+<gloss>at the beginning (of the month, year, etc.)</gloss>
@@ -28,0 +31 @@
+<s_inf>often kana only</s_inf>

Added noun meaning and split senses.

@@ -12,0 +13 @@
+<ke_inf>&iK;</ke_inf>
@@ -24,0 +26,5 @@
+<sense>
+<stagk>始めに</stagk>
+<pos>&n;</pos>
+<gloss>Preface</gloss>
+</sense>