Thoughts on Newton and Einstein

An old conversation came back to my mind a few days ago about whether Newton's theory of gravity and mechanics is valid or was falsified by Einstein's theory and related evidence. Through an analogy, I'll explore reasons why I see Newton's theory as falsified despite still being useful.

I am trying to discover how a certain natural phenomena works. Only mother nature knows the true answer, we'll call that nature's law (NL). Although I haven't figured it out yet, nature has (NL): y = e^x. Note that NL could also be written out in the form of a Taylor series.

Based on my experiments, I have formulated a theory (T1).
(T1) y = 1 + x/A!, where A=1 (±0.0001)

In all my experiments, I was limited by the precision of my tools in measuring values of x and y. Over time, I have been able to infer A with more precision. I expect this narrowing to continue.
Although all the evidence so far conformed closely to T1 in my experimental settings (x close to zero), I later discovered some discrepancy for larger values of x.
Based on that evidence, I have formulated a new theory (T2).
(T2) y = 1 + x/A! + x^2/B! + x^3/C!, where A=1 (±0.0001), B=2 (±0.0001) and C=3 (±0.0001)

Since T2 adds some second and third order factors (compared to T1), it explains why T1 was my best theory for the longest time.

In addition to those two theories, let me enumerate a few variants to consider:
(T1a) y = 1 + x/A!, where A=1 (±0.0001) and x is close to zero
(T1b) y = 1 + x/A! ±e, where A=1 (±0.0001)
(T1c) y = 1 + x/A! ±e, where A=1 (±0.0001) and x is close to zero

We are now ready for the main questions:
1) Which of those theories (if any) is analogous to Newton's theory? Which of those (if any) is analogous to Einstein's?
2) All these theories are useful in practice, but how would you qualify each of those theories: valid, true, falsified, false, ...?

Here are my thoughts:
1) What's the analogy?
T1 is analogous to Newton's original formulation.
T1a is analogous to its modern formulation.
T1b and T1c are not analogous to neither theory. Neither theory includes an epsilon or "alsmot equal". As a side note, it's important to conceptually distinguish such epsilon (an uncertainty in the theory, really to hide the fact that we have not figured out the missing terms) from measurement uncertainty. Even with super precise measurement, the epsilon would remain in T1b and T1c.
T2 would be analogous to Einstein's theory.

The core of this analogy is that the 1st order Taylor series (T1) approximates to the exponential (nature's law) and the 3rd order Taylor series (T2) when x is small, just like Newton's mechanics is a fine approximation of nature and Einstein's theory when speeds are non-relativistic (v << c).

2) Which theories are falsified and why?
T1 is false/falsified, because it is inconsistent with the evidence at larger values of x.
T2 is considered valid at all ranges of x, to the best of our knowledge at this point.
If T2 is the best of our knowledge, then T1a is falsified. They cannot both be true simultaneously. The missing terms in T1a are camouflaged by measurement imprecision, but we already know by continuity to larger values of x that as our measurement gets better, the missing terms will become apparent. So even if T1a was not invalidated by evidence (with x close to zero), it is invalidated by our broader knowledge.

Parting thought
What I tried to illustrate with this analogy is that even if the evidence in limited conditions could still be compatible with Newton's theory, that theory was falsified by the discovery evidence outside that range.
In practice, we don't even have to resort to this reasoning to falsify Newton, as relativistic effects become apparent without going that fast, if you are precise enough (GPS satellites). As you slow down even more, those effect require even greater precision to detect, but they are there nonetheless.

HT Bertrand Le Roy (who prompted this reflection with an interesting Thanksgiving conversation) and Kenny Herrington and Hank Hoek for their kind feedback.