Definitions of an algorithm running in polynomial time and in strongly polynomial time
Before formal definitions, consider what the "strongly/weakly" classification aims to distinguish.

First, consider running either one on a Turing Machine. Both will run in a number of steps polynomial in the length of binary-encoded input. Consequently, the number of arithmetic operations performed by both would have to be polynomial in the length of binary-encoded input. Hence, for both Turing Machine running time would grow polynomially as the number of input values or their magnitudes grows. To emphasize the latter, note that even the strong one will take more TM steps on larger magnitudes (it needs to at least read the extra bits). Under no circumstance does one become exponential (as is the case with the unrelated pseudo-polynomial time). With a Turing machine alone, a fundamental difference doesn't seem detectable.

Now consider running each on an arithmetic machine where any operation is an arithmetic operation on whole numbers and arithmetic operations are $O(1)$. As you increase the magnitudes of the input numbers, the length of the binary-encoded input grows, and the running time of the weak algorithm would grow, however the running time of the strong algorithm would not change, because can be bound by the number of input numbers, which you are not changing (e.g. matrix multiplication vs. Euclid's GCD).

The set of algorithms that run in number of arithmetic operations polynomial in the number of input numbers is well-defined, but overlaps with the class of algorithms that take number of TM steps exponential in the length of binary-encoded input (see examples). Hence, for this set, the properties in the second paragraph would not hold. To exclude the unwanted intersection, we add a condition for a polynomial TM space [*]. 

In [1] this is stated in two ways:


The algorithm runs in strongly polynomial time if the algorithm is a polynmomial space algorithm and performs a number of elementary arithmetic operations which is bounded by a polynomial in the number of input numbers.
A polynomial algorithm is a polynomial space algorithm (in our standard Turing machine model) and polynomial time algorithm in the arithmetic model (see this question for a clarification).


This sub-classification of what are already "efficient" algorithms might be  be useful since physical computers resemble arithmetic machines more than Turing machines, so the sub-class might identify the algorithm that would run faster on a physical machine if their polynomial bounds in length of binary-encoded input are otherwise comparable (which is not the case in e.g. strong $O(n^3)$ vs. weak $O(n^2)$)

[*] The second condition is stated everywhere as polynomial space, whereas requiring polynomial time makes more sense to me. The former is more inclusive, but strange. Are there strongly-polynomial algorithms that take more than polynomial time? Note that repeated-squaring example takes neither polynomial time nor polynomial space. 

[1] Grötschel, Martin; László Lovász, Alexander Schrijver (1988). "Complexity, Oracles, and Numerical Computation". Geometric Algorithms and Combinatorial Optimization. Springer. ISBN 0-387-13624-X.