American English - the differences

I'm trying to understand when American English misses out the 'on' in, for example:

"In essence, Biden's conduct Wednesday..." vs "...the first full day of Biden's term on Thursday..."

Taken from this CNN article.

It seems a bit inconsistent to me, or are there specific rules?

In British English, we always 'on' for days of the week prepositions.

Quote from Rice

I believe it’s more common to use “on,” but skipping it is acceptable shorthand.

Shorthand in news articles?