precision@k and recall@k
A quick answer.
Note: Checking the references I could access fully, there are no discrepancies between the definitions as long as the terms are translated correctly.
Some definitions:
Relevant items: Are items the user(s) have themselves established as relevant with their actions.
Recommended items: Are items which the recomendation system predicts will be relevant to the user(s).
Concerning your example:

Let’s understand the definitions of recall@k and precision@k, assume
we are providing $5$ recommendations in this order — 1 0 1 0 1, where 1
represents relevant and 0 irrelevant. So the precision@k at different
values of k will be precision@3 is $2/3$, precision@4 is $2/4$, and
precision@5 is $3/5$. The recall@k would be, recall@3 is $2/3$, recall@4
is $2/3$, and recall@5 is $3/3$.