Learn English – “Became” vs “become”

1. Use the present if what Peter was trying to convey is about something being true in general or persistently (regardless of whether that something is in fact true in general...).

   This example is clearer, I think: Peter said the sky is blue -- vs -- Peter said the western sky was a particularly bright orange when the sun hit the horizon that evening.

2. Yes, you can remove that "that". But the sentence can be easier to understand, esp. by some non-native English speakers, if you leave it in. And esp. since you do not use "the formula B" here, you can say "... that formula A is based on formula B."

Take time and tense out of the question, to see what it means to make a general (and atemporal) statement (which one can make at any point in time). And no, quotation need not be involved either.

• Water is a colorless, transparent liquid. [general statement about the nature of water]

• The water in my glass is cloudy and brown. [specific statement about this water]

• Peter explained that water is a colorless, transparent liquid.

• Peter told the waiter that the water in his glass was cloudy and brown.

It is correct to say either of these:

• Ptolemy claimed that the Earth is at the center of the [solar] system.
• Ptolemy claimed that the Earth was at the center of the [solar] system.

They both say that Ptolemy made a general statement. Neither is incorrect. The former emphasizes the generality of what was claimed (including its abstraction from time). The latter emphasizes the claim as historical claim: the time-limited nature of its meaning.

IOW, it all depends on what you are really trying to say. This is about logic and meaning as much as it is about language (wording).

And FWIW, I agree 100% with this post about the same topic. Both can be correct, and there can be a slightly different connotation (emphasis) if you use one or the other. Neither is "awful".