Pardon me for this somehow lengthy response.
The episodes 1 and 7 of Steins;Gate tell a lot.
In the first half of episode 1, Okabe received a video mail from the future. It was received on 28 Jul 2010.
The main content of the mail, with the attached video file, in episode 1:
The video included in the mail, still locked, not playable (static), in episode 1:
The video, unlocked, playable, in episode 23:
This video mail definitely shows the received date, not the sent date.
In the same episode 1, we have the first D-mail which is what switched the world line from that of Beta to Alpha, sent on 28 Jul 2010 and received on 23 Jul 2010.
The mail received in Daru's phone:
The current date, which is consistent with the received date of the video mail above (28 Jul 2010):
Daru's received mail also shows the received date.
At the beginning of episode 7, they made a D-mail as an experiment which was to be sent on 3 Aug 2010 and received on 29 Jul 2010.
The timestamp shown at the very beginning of episode 7:
The contents of the D-mail sent:
This mail, too, uses the received date.
There are a few other time travel experiments but that's pretty much it for the anime, we can also consult the visual novel, its source material.
Unfortunately, Okabe's phone in the visual novel does show the receiving date, but not the year. The year has to be implied to be the current year unless otherwise specified by the characters reading the mail or the narration. Also, only the receiving date is what is normally shown.
What a mail normally looks like in the visual novel, with only the received date shown:
The above screen shows the LOTO SIX D-mail which was to be sent on 3 Aug 2010 and received on 27 Jul 2010. Just like the anime, it also only shows the received date.
A rare occasion where both the received and sent dates are shown:
When receiving D-mails, which date is supposed to be used? The sending world line's date? Or the receiving world line's date? Normally, this doesn't matter since the send and receive dates will be almost the same for electronic communication.
With the above in mind, it is safe to assume that the date on the field ”受領日時”/"Date and time received" should always be the date the mail was received. In other words, the mail "Turn on the TV."/"Turn on the news..." in the episode 23 of Steins;Gate has an incorrect date, and should have the year
2010, not 2025. Steins;Gate 0 episode 23 seems to correct it.
Is this discrepancy intentional? Or is it a goof? If it is intentional, is there yet another world line change hidden underneath that we don't know yet? (The ending of Steins;Gate 0 left plenty of room for a 3rd season.)
It is most likely a mistake. Steins;Gate the anime is known to have its fair share of inconsistencies when it comes to details. As for Steins;Gate 0, it was meant to expand on the original, especially its beginning and ending, and explore the Beta world line(s). Before Steins;Gate 0, the Beta world line wasn't really explored because the original focused on the Alpha for the most part.