It is hard to have message authenticity without integrity. To authenticate the message you need to know what message is being authenticated. If you could change the message the authentication tag should become invalid. Message authenticity means that you can establish that the message originated from a trusted entity. For this reason message authenticity often implies integrity.
This doesn't go the other way around because of checksums. Checksums can be used to protect integrity of a message. Checksums however do not protect against active attacks as anybody is able to calculate a checksum. If you need to protect against active attacks then the message is usually authenticated as well.
I'm not sure that "data authenticity means that the initial message sender is who he/she claims to be" is a correct definition. That seems the definition for entity authentication. The data may not be an identifying aspect for the entity. I would rather say that message authenticity proves that the message originated from a specific entity. In that case - as explained in the first section - integrity is probably implied.
As an example, say we find a bunch of paintings all bearing the same signature. We however do not know who created the paintings. All we know is that there is some unknown painter that painted the messages - the identity of the painter is not really established. What we do know however is that all the paintings belong to a single painter. That says quite a lot about the paintings themselves.
If you look at entity authentication then no explicit message needs to be involved, so data integrity may not be required. So there you are: authentication without integrity.
Authenticity and integrity are different concepts, even if used in the context of data/messages. It doesn't make sense to mix those concepts even if one implies the other. So I try to write "message authentication and integrity" instead of just "message authenticity" or "message integrity".
In text you have to distinguish between message and entity authentication. If you have message authentication you know that integrity is very likely implied. This is not the case for entity authentication. So you do need to look at context to see if integrity is implied.