In my grammar book, it says that てくる is used in two ways:
1) It is used with verbs that indicate a process that takes time to complete (e.g. 太る ---> 太ってきた = started to gain weight).
What is the difference between this and the 〜masu stem form of a verb + 〜始める? Don't they both mean begin?
2) It is used with a non-punctual verb (I don't know what that is) to indicate a continuation of sth. up to a current point in time e.g. 今までたくさん本を読んできました.
I don't understand the difference between using 読んできました and 読んでいました. In the above sentence, 読みました (read/have read) seems more fitting than either of them (Up to now I have read quite a few books).