Say that H = hypothesis, O = observation. To compare the cases clearly, let's draw a Venn diagram of the possible outcomes:
│ │B │ │
There's a box for H which intersects with the box for O. I've labeled the four areas: A = H \ O, C = O \ H, B = H ∩ O, D = (H ∪ O)ᶜ.
Notation: \ is set difference; H \ O means "the parts of H that are not also in O." ᶜ is set complement; (H ∪ O)ᶜ means "all the possible outcomes not in H and not in O."
If H is highly likely given O, this means that P(H|O) = P(B) / P(B ∪ C) is large (close to 1).
If Peter Lipton's condition, "had H been false, O would have been different," holds, this means that P(O|Hᶜ) is low (close to 0). So this is saying that P(C) / P(C ∪ D) is small (close to 0).
Now, to see the difference, consider if the outer region D is very high probability (a very large region in the Venn diagram) which also means that H and O are, a priori, both unlikely. In this case, Lipton's condition P(C) / P(C ∪ D) is always going to be small, because the size of region D dominates the size of region C.
But increasing the size of region D has no effect on the condition that "the conclusion is probably true, given the premises"; P(H|O) is unchanged.
So, Lipton's condition as you've stated it would call an inductive inference strong, whenever both hypothesis and the observation are a priori unlikely, regardless of whether the hypothesis is related to the observation. This is probably unintended behavior. I would suggest that Lipton's condition is mistaken, or perhaps you've misinterpreted it.