核心问题
（1）The telegram sent by my sister has been received.
(2) Sending messages by telegraph will be more popular.
telegram 与 telegraph 都作“电报”解, 它们有无区别呢?
词典解答
telegram 既可表示“一份电报”，亦可表示作为通讯手段的“电报”：
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He received a telegram saying that his mother had been in a critical condition. 他接到他母亲病况危急的电报。
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We sent them a message of congratulations by telegram. 我们给他们发了一封贺电。
telegraph 主要是指作为通讯方式的电报,如你所提到的句(2):
The news came by telegraph. 这消息是由电报送来的。
此外，telegraph 还作“电报机”解。