Correct

开始 vs 开头

/ kāishǐ vs kāitóu /
Strokes

Definition

开始 as a Verb or Noun, and 开头 just as a Noun, which both means [begin; start], indicate the initial stage of development.

Example

开始 and 开头 have different stresses, 开始 is just a beginning stage, while 开头 means "the beginning of an obvious process". 开头 is more colloquial.
It is not suitable to use 开头 in example ①. If 开始 is used in example ②, one ought to say 开始的时候 or 开始的部分.
1
Hǎo
de
kāishǐ
开始
shì
chénggōng
成功
de
bàn
一半
  

A good beginning is half done.

2
shì
故事
de
kāitóu
开头
hái
tǐng
yǒusi
有意思
  
,
hòulái
后来
jiù
xíng
不行
le
  

The beginning of the story is interesting,but the rest is not so.

开始 is also a verb, meaning "to start from the beginning or from a certain point". 开头 is also a verb-object phrase, meaning "to start from the beginning", which is very colloquial.
1) Following is a comparison of the verb 开始 and the verb-object phrase 开头.

2) 开头 cannot be used either in example ③ or example ④.

3) 开始 cannot be used in examples ⑤ and ⑥, since 开始 only means "to start from the beginning", but 开头 stresses "how a process begins from the very beginning". Especially, here 开了个头 is rather special.
3
Xiànzài
现在
kāishǐ
开始
shàng
上课
  

Now class be-gins.

4
huì
大会
kāishǐ
开始
le
  

Now the meeting begins.

5
Zhè
cháng
tǎolùn
讨论
gāng
kāi
le
tóu
个头
  

This is just the beginning of the discussion.

6
zhīdào
知道
zhè
piān
wénzhāng
文章
zěnme
怎么
kāitóu
开头
hǎo
  

I don't know how to begin the article.

开始 and 开头 are interchangeable as there is obviously a process (学期 and 讲故事).
Either word can mean "the start of the process", but 开头 is more obvious in this aspect.
7
Zhè
xué
学期
gāng
kāishǐ
开始
  
(
kāitóu
开头
  
)
  
,
háiyǒu
还有
sān
duō
yuè
ne
  

This is just the beginning of the term, and there are over three months left.

8
jiǎng
de
shì
故事
cái
kāishǐ
开始
  
(
kāitóu
开头
  
)
  
,
mànmàn
慢慢
tīng
  
!

He just starts telling his story, and you'd better follow it with patience.