Here's an example of a part (17836) that shows this text molded on the underside:
When did bricks start to have the LEGO Group text engraved on them?
And what was the first set to use these?
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It appears that this text was introduced sometime around 2010.
It's hard to determine a definitive answer here, because not all new parts include this text (nor even the LEGO logo). This is likely often due to space constraints, but there may be other considerations as well.
The 8x16 tile from 2004 does not have this text despite being large enough:
The 2006 2x3 tile with clips is also missing this text:
However, some variants of the 2x4 tile introduced in 2010 do have this:
So it appears to have been added around this time.
The 2x3 tiles introduced in 2016 also have this text.