Which one we should use when we're sending emails to customers to inform them about news or other information?
any other formal or informal suggestion?
As a native speaker, my preference is "for your information." The other feels stilted, and to me feels like a non-native construction. In fact, if I were reading your message, and I saw "For your notification," I would automatically assume the message was written by a non-native English speaker.
It's worth noting that the phrase "for your information" is sometimes used rhetorically as a defensive statement when one person is trying to respond to criticism, or trying to establish credibility in a defensive way. Consider, "For your information, I DO have a PHD in Warp Field Dynamics, and I have 21 years of experience working with star ship engines." When you hear this, you can tell the person is being defensive, but when you are reading a message, it is harder to tell intent.
So, if I were writing the message, and I wanted to use "For Your Information" as a heading, I think that is fine. If I were wanting to include it in a sentence, I would probably couch it in a less potentially-confrontational way like this: "We wanted to let you know about some changes that have happened..." or "We wanted to inform you about some changes...". This has a friendlier tone than "For your information, we are making changes...".
notification by definition implies a notice:
- The act or an instance of notifying.
- Something, such as a letter, by which notice is given.
For example, if you don't pay your electric bill, you will receive a notice/notification that the power will be turned off unless you pay.
Since you are providing information, use for your information.
However, notification might apply if the information affects the status of products or services already in-process or completed:
This notification was sent to advise you regarding a recall of the item you recently purchased.