伝わる and 伝える: communicating and getting the meaning across
Compare the act of conveying information with its meaning getting across, and distinguish the message, recipient, and source.
The message and its recipient
先生に欠席の理由を伝えました。 says that someone told the teacher the reason for their absence. 伝える, read つたえる, marks the recipient with に and the content with を.
If the meeting time has changed, one can say 会議は三時からだと伝えてください. The clause before と is the message; distinguish the request to relay it from the time quoted within it.
Whether the meaning got across
私の言いたいことが相手に伝わりませんでした。 means that what the speaker wanted to say did not get across to the other person. With 伝わる, read つたわる, the intended message is the subject.
説明はしましたが、意図が十分に伝わったか心配です。 says that the speaker explained but worries whether the intention came across fully. Speaking and being understood are not the same result.
How news spreads
その知らせは友人から家族に伝わりました。 says that the news reached the family through a friend. から marks its source and に marks its recipient.
感謝の気持ちを手紙で伝えました。 means that someone expressed gratitude in a letter. Feelings as well as factual information can be conveyed, and 手紙で gives the medium.
Does 伝えました guarantee that the recipient fully understood the intended meaning?
Show the answer and why
Not necessarily. It describes the act of conveying something. Whether it was adequately understood can be checked separately with 意図が伝わったか.