(One example source on what I mean, you'll have to take my word they are the from the same thing or look it up yourself unfortunately, I couldn't make an image with both of them)
So I'm not sure if this response follows the rules, but the situation is somewhat unique because modern sources appear to have made a technical error. The moderators can remove this post or let me know if it doesn't follow the rules.
The Oxford Handbook of Christmas, 2020 Edited by Timothy Larsen reports the same article you've screen-shotted as new evidence of earlier Christmas cards. Here is a screenshot of the relevant page mentioning this article from 1829:
I think that all three of us (you, Timothy Larsen and myself) are all finding this via the British Newspaper Archive, which does in fact show this as an article from 1829:
Searchable indexes are amazing for this kind of research, but they are only as good as they are accurate. If you look at the text scanned on the BNA as page 6 of this edition of this newspaper, it's clearly a clipping that has been separated from the rest of the paper at some time and not a full page of text with a date on it:
So it's entirely possible that this loose clipping was mixed up along the way and re-filed with the 1829 edition and is thus mis-attributed and not really from 1829. After all, the actual clipping itself has no date on it.
We might determine its actual date by looking at other names mentioned in the same clipping to see if this chronology makes sense. If fact, the first thing in that page of clippings is a reference to the engagement:
Some quick googling will lead you to the fact that these two people were married in *1931*, not 1829 or 1831. So the most likely explanation here is that a technical error occurred at the British Newspaper Archive and a ~1929 clipping was misfiled under the 1829 edition. But based on the context of the clipping, there's no reason to think that this is actual evidence. In fact, several posts this year that mention it seem to be incorrect.
Update: Professor Larsen replied and said he was swayed by this evidence. I invited him to post here if he is inclined.
Update:
Reply from Professor Larsen, posted with permission:
"I am truly grateful for your thorough and convincing investigation – and for taking the time to present your research to me so clearly. I announced this alleged find in a letter to the Times Literary Supplement last year and multiple scholars wrote to me about and some tried to hunt for it. I wrote to the local county archives myself. In short, no one figured out what you did. In an age – at least here in America – when people would rather believe what they wish was true than weigh evidence it is important to me to say clearly that I have been persuaded by your argument (though I have not yet looked at the images you sent) and will give up my pet theory and alleged find. Many thanks for your work. Best wishes, Tim"
Last Update: Professor Larsen send this message to the Times Literary Supplement. It's really cool that a random question on AskHistorians changed our understanding of history in its own tiny, tiny way:
From: Tim Larsen
Subject: Christmas is still Victorian
Christmas is still Victorian
Sir, - In a letter in the 24 & 31 August 2018 issue (‘Christmas gets earlier’), I announced a card mentioned in the Hampshire Chronicle in 1829 as a new contender for the earliest commercial Christmas card. The book I was then editing, The Oxford Handbook of Christmas, has recently been published which has led to renewed interest in this claim. A crowd-sourcing effort – with the initial, decisive breakthrough having been made by (my name here) – has now refuted this claim. It seems that a clipping without a date on it – probably from 1929 – has been misfiled and then falsely dated and encoded in the British Newspaper Archive. Henry Cole continues to be king and, in the spirit realm, can be a particularly merry old soul this Christmas. I, on the other hand, will go back to telling my students curmudgeonly that there is no substitute for direct, archival research.
TIMOTHY LARSEN
Wheaton College, Wheaton, Illinois.
Wow. That is incredible. I will use this as an example to my pupils about the importance of verifying your sources. Awesome work!