I recently out of curiosity came across this paper summarizing Tate's thesis which, in its introductory section, claims that, given a number field $K/\mathbb{Q}$, a prime ideal $\wp$ of $O_K$ is ramified if, when it lies over the prime $p$ of $\mathbb{Q}$, $\wp^2\cap\mathbb{Z}\neq(p).$ But this clashes with other definitions of ramification I've seen, because I thought it was usually used to describe the primes of $\mathbb{Q}$, or in general the base field of the extension, rather than $K$.
Also, all other definitions I've seen seem to imply that, for example, the prime 2 ramifies in $\mathbb{Q}(i)/\mathbb{Q}$, as $(1+i)^2=(2)$, so that the prime ideal $(1+i)$ appears as a repeated factor in the factorization. However, by the definition given in this paper and the fact that $(1+i)^2=(2)$, the fact that $(2)\cap\mathbb{Z}=(2)$ would seem to imply that $(1+i)$ is not ramified in this situation.
I'm somewhat of a beginner to algebraic number theory, and I figure I must be making some sort of mistake or misinterpreting the definition. Are these just two definitions of differing concepts, or have I made an error?