Persist username across login/register form toggle

[Copilot]

When switching between the login and register forms, the username field was reset because each form managed its own local name state — unmounting one form discarded the value before the other mounted.

Changes

• Lift name state to AuthScreen: AuthScreen now owns const [name, setName] = useState("") and passes it down as props, keeping the value alive across form switches.
• LoginUserForm / RegisterUserForm: Replaced local name state with name: string / setName props.

// AuthScreen.tsx
const [name, setName] = useState("");

// passed to both forms:
<LoginUserForm onLogin={...} name={name} setName={setName} />
<RegisterUserForm onRegister={...} name={name} setName={setName} />

Original prompt

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This section details on the original issue you should resolve

<issue_title>Persist user name when switching from login to register or vice versa</issue_title>
<issue_description>The user name input field should not be cleared when switching between the two tabs (typically using the link below the login/register form)</issue_description>

Comments on the Issue (you are @copilot in this section)

• Fixes Persist user name when switching from login to register or vice versa #49

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