Many in the technology industry collectively agree that the title 'Product Manager' is not well defined and I have seen this confusion for years. I will not attempt in defining the term because it, quite practically, differs from company to company. How technical a Product Manager should be or can get also differs from company to company. In smaller companies, the roles and responsibilities are wide spread however in larger organizations, some of the functions are distributed among different people, team or even departments.
In software based product companies, in small companies where there are not enough resources to split the role into product management and project management, it is typical that one person is responsible for both actions. However in medium sized and large companies, these responsibilities are divided among teams or people. Here is where it becomes difficult to understand the responsibilities and division of labor.
I have been in such positions several times in my career (both as a Project Manager and as a Product Manager) and this is how I try to limit or extend myself. The way I managed changed depending upon the situations but here is my recommendation:
Whenever a task falls on you, or you try to pick up - ask yourself this question "does my decision affect the user or is it transparent". If it affects the user (or the product), then it is definitely Product Manager's job. If it is transparent to user, then it probably isn't Product Manager's job (example, it probably doesn't affect user if a code is written in one logic or another as long as the work is done; or it doesn't affect end user which developer is going to work on it and when developer finishes and hands over to testing).
Product Manager's job is ideally related to traditional 4ps such as Product, Price, Promotion. Project manager's job is ideally execution and operation to achieve the target keeping in mind three pillars of Quality, Time, and Cost.
All said and done, Product Manager's "job" is to ship right products out of the door on time..and she will have to do anything (whether or not someone else is technically required to do that job) that will help to achieve that including knocking on developer's doors or sweeping the floor if need be.