As @Julian said, you will do well with the space $H^1$ and test functions are the keyword. But I would like to emphasize this point, which to my opinion might close the gap here.

You need to pay attention to your test functions: If you say that different BC give rise to the same weak formulation then this is "true" and false, but more precisely it's false. The weak formulations might look the same but they are not. This is best explained by reminding of two essential properties of a weak formulation:

- A regular enough weak solution satisfies the original problem classically
- A classical solution to the original problem satisfies the weak formulation.

So if you say two different BC give the same weak formulation then this would mean that at regular points of their solutions these solutions obey different PDE classically......

The point is that in the weak formulations for different BC you employ different test functions. Otherwise how do you show that a regular enough weak solution obeys the classical problem? If your test functions all vanish at the boundary, then you could not conclude that at points where they are regular they obey a Neumann boundary condition classically. So you need test functions in $H^1 \cap C^{\infty}$ for example. (Maybe try to prove that assuming you have a regular weak solution at hand it also satisfies the classical problem. In this process you will see why the choice of the test functions is essential)

Shortly: To ask for $u$ satisfying equation xyz for all test functions $\phi \in H^1_0$ is a different problem (with different solutions) then searching for $u$ satisfying equation xyz for all test functions $\phi \in H^1$.

Addendum: I would like to emphasize that, even if you know nothing about the regularity of your weak solution(except the weak regularity provided by your weak solution functional setting, like $L^2_tH^1_x \cap L^{\infty}_tL^2_x$ for instance, depending on your operator $L$) you have perfectly different weak formulations for different BC. The only question is whether your weak formulation is really a weak formulation for your classical problem, and for checking that you only hypothetically need to say, okay what if my weak solution is regular, does it then obey the classical problem or not. So you don't need to study additional regularity questions for justifying and solving your weak problem.