I'm still fuzzy on the distinction you are trying to make between "writing algorithms" and "writing code". If someone told me they were "writing an algorithm", I'd understand it as them trying to come up with a new algorithm, either on paper or in code.
If you're given the outline for an algorithm, and asked to write code to implement it, is that writing algorithms or writing code?
This is just me, but in the two examples the first one is algorithm and the second is code. The difference being that in the first one he starts to explain the sequence he wants but apparently expects you to complete the sequence to code the function. Granted he does give you enough of the sequence that you can probably see the pattern. The second he basically tells you the equation he wants to see in code.
The first one requires an ability, even if minor, beyond just coding while the second is all about coding. I also find it interesting that he says the second version was more difficult for his applicants than the first. Maybe instead of "algorithm" it's more about pattern recognition that leads to minor math skills to then coding.
If you're given the outline for an algorithm, and asked to write code to implement it, is that writing algorithms or writing code?