Are weakly polynomial time algorithms truly polynomial?
As Wikipedia says, there are two conditions for strongly polynomial time algorithms:


  
  the number of operations in the arithmetic model of computation is bounded by a polynomial in the number of integers in the input instance; and
  the space used by the algorithm is bounded by a polynomial in the size of the input.
  


I will simply call these two conditions as condition 1 and condition 2 in this answer.




  The algorithm is already polynomial time in the arithmetic model, so as long as its space is bounded in the TM model then the rest of the algorithm is still polynomial time in the TM model. Think of it like this, when converting from the arithmetic model to the TM model, the only thing that can go wrong is the space. The numbers you produce need to be representable in a number of bits that is polynomial in the number of bits in the input. So long as that holds, the rest of the algorithm is still polynomial time in the TM model.


This paragraphs is describing why condition 2 is necessary for strongly polynomial time algorithm, it is not stating the difference between strongly and weakly polynomial time algorithms. In fact, condition 2 is also necessary for weakly polynomial time algorithm, and the example algorithm that computes $2^{2^n}$ given $2^n$ is NOT weakly polynomial (and is even not a polynomial time algorithm).

In fact, it is condition 1 that describes the difference between strongly and weakly polynomial time algorithms. Note the definition of strongly polynomial time algorithms is equivalent to


  
  condition 1; and
  the algorithm is a polynomial time algorithm (in the TM model)
  


While the definition of weakly polynomial time algorithms is equivalent to 


  
  condition 1 is not satisfied; and
  the algorithm is a polynomial time algorithm (in the TM model)
  


It is more clear to see the difference between strongly and weakly polynomial time algorithms with these two redefinitions.



This post is wrong to suggest the difference is weakly polynomial time algorithms don't run in polynomial time on a TM. I have added a comment on that post addressing this issue.