If Einstein’s theory is true then light rays passing close to the sun are deflected. Careful experiment reveals that light rays passing close to the sun are deflected. Therefore Einstein’s theory is correct

It helps to reduce the argument to its logical form, then substitute.

If x, then y.
y.
Therefore, x.

This is a clear Affirming the Consequent fallacy as you point out. Remember that fallacies don't mean that the conclusion isn't true; they simply mean that the argument cannot logically/rationally justify the conclusion. In other words, a fallacious argument does not make the conclusion wrong (see Argument from Fallacy ).

If Socrates is a man, then he is mortal.
Socrates is moral.
Therefore, he is a man.

It sounds like affirming the consequent , but since we're dealing with a scientific theory, you can't look at it too simplistically.

Firstly, if a theory makes predictions, and there is evidence for these, then that counts as evidence for the theory. Inversely if there is no evidence where it should exist, this is negative evidence, or evidence against the theory.

However, you're going to need more than one instance - proving Einstein's "theory" (whichever one it is) probably won't be done with one piece of evidence. This is because the outcome ("rays passing close to the sun") could be explained by other theories.

E.g.

"If you are right, then my car is in the garage."

"My car is in the garage, so you are right." (about what?)

If "right" refers to believing the car is in the garage, then this is a correct conclusion. Another theory to explain my car being in the garage might exist, but it just might not be as plausible as the first.

Both Einstein and Newton predicted a shift, but Einstein's was more accurate.  The Eddington experiment was a proper falsification test, in that was measuring an event that might have shown a)no shift b)Newtonian shift c) Einsteinian shift. 

The physicist/astronomers took measurements by methods all agreed would observe the light, and after the observation, the shift was shown to match prediction (c).

The better way to characterize the experiment is

If Einstein’s theory is false then light rays passing close to the sun will not be deflected as Einstein predicts.

The rest, as they say, is history.

There were no fallacies the way the falsification trial was enabled.  The wording of the instant question is not what they did or how they thought.

It's a matter of presentation...

Let's fix things a bit.

1.Light rays passing close to the sun are deflected. (This is a claim that has been verified)

2. Einstein theory of light diffraction states that light rays passing close to the sun are deflected.

3. Therefore, Einstein's assertion was/is correct.