Where you store the type information and when you do the type check is a separate question from whether you do the type conversions automatically or not.
I think a more interesting question is typecasts, like happens in languages like Java and C#. These languages are nominally statically typed, but they retains some type information at run-time, so that you can perform run-time type conversions, which requires run-time type checking. Which is the defining feature of dynamic typing.
C# is a little bit more straightforward about being a hybrid static/dynamic language, with its reified generics and dynamic references. But teasing out the details of where, how, and the extent to which Java is statically or dynamically typed would make a decent topic for a master's thesis.
It also hints at a deeper thing that one must be mindful of: static/dynamic and strong/weak are not binary categories. They're not even the extremes of two binary scales. They are somewhat vague descriptions that are meant to serve as useful shorthands for certain sets of choices that one must make when designing a language's type discipline.
But the fact that they're not cut-and-dry terms does not mean that they're meaningless. It just means that one must disabuse oneself of the notion that they're cut-and-dry before one can have a conversation about type discipline that goes beyond a certain level of detail.
