The Japanese example screenshots show a Brilliant Diamond game and its Pikachu with Tail Whip, Growl, Quick Attack, and Thunder Wave as the latest moves a Pikachu would be generated with in both SWSH and BDSP since they use the same learnset. The equivalent English screenshots show a Pikachu in Violet with Nasty Plot, Play Nice, Nuzzle, and Thunder Wave, which is made out of moves it could have relearned. Notably both have the Galar origin mark and originate from SWSH. As we'll see later this is a reference to the Level 5 Pikachu you get from starting up HOME, though neither have preserved its original moveset.
There is also a Lv. 25 Pikachu shown in Mobile HOME with the Alolan origin mark and Thunder Wave/Spark/Electro Ball/Feint. Notably the game-specific details feature includes the Bank profile, which is a new feature for the mobile version so you might be able to look at your legacy moves in some form. What's kind of odd here is that the Bank icon seems to be showing that it's being viewed under its Bank stats, but Pikachu only learned Spark at level 26 in Gen 7. There are also options for SV and SWSH views and no height/weight stats listed.
On the Japanese side, this Pikachu is probably the same one used in the BDSP example with the same moves and BDSP and SWSH POVs available, and SV and PLA greyed out unlike the English version. Based on the OT and TID being HOME and 200212, this is actually supposed to be the HOME gift Pikachu you get from starting up HOME, which would normally have Thunder Shock, Tail Whip, Growl, and Quick Attack. It even has the same stats with all 20 IVs!
There are equivalent screenshots for the Level 5 Pikachu in Switch HOME as well, though with no Bank option here. The Pikachu have access to SWSH and their respective example games shown.
EDIT: I almost forgot to check the Chinese and Korean sites, though the Korean site hasn't updated yet.
The Chinese versions show a SWSH native Pikachu with only SWSH options, but also showing the other Gen 8 games greyed out. They have the HOME OT but the wrong ID number and it doesn't even have all 20 IVs like the real HOME gift Pikachu so this is just a shoddy reproduction.
Their HOME Move example still shows a BDSP perspective though like the Japanese versions. However, the Japanese and English HOME move Pikachu shared the same stats while this Pikachu actually matches the Chinese phone example's stats and has no blue markings unlike the other languages.
tl;dr: There are multiple different Pikachu shown in the examples across languages.
The Japanese HOME side consistently uses a Level 5 Pikachu with a SWSH origin that has been transferred to Brilliant Diamond and is implied to be the HOME gift.
The English/other western language HOME sides mainly show a Level 5 Pikachu with a SWSH origin transferred to Violet, and is also implied to be the HOME gift.
The English mobile HOME example shows a completely different Level 25 Pikachu from Alola.
The Chinese side also shows a Level 5 Pikachu with a SWSH origin into BD like the Japanese side, but theirs has different stats and no markings. It has the HOME OT but no other signs of it being the real HOME gift.
Speaking of the Chinese side, this is another source that supports the "moves it learns in the game you're entering" interpretation since it uses the term 送去 which means "sent to" compared to the Japanese woridng used which could apparently mean both before or ahead.
https://www.pokemon.co.jp/ex/pokemonhome/sc/features/03/
Overall I think the English side's wording is pretty unclear especially with how they specify (some moves excluded) only on BDSP, and it's hard to say what will actually happen in practice, but I would rather keep my expectations low and be surprised if it does actually happen that you can mix and match your movesets across games.
Also this tera type thing could be nice to get something like a Water tera Sap Sipper Marill more easily, because as of now you either have to change tera types from a raid or bred Azurill because Azurill can't spawn with Water tera from breeding, or Ability Patch a wild Marill with water tera, or find a natural water type Marill raid with HA. Another example is Ghost tera Decidueye or really anything that gets a new type on evolution that you wouldn't be able to tera spawn naturally.