Use the current context instead of the browser context for backup result API

[bbondy]

• Tor window should use Tor
• Proxy should use a proxy https://developer.chrome.com/docs/extensions/reference/proxy/

Test Plan

• https://github.com/brave/internal/issues/766

[kjozwiak]

Verification PASSED on macOS 11.2.3 x64 using the following build:

Brave | 1.23.73 Chromium: 90.0.4430.85 (Official Build) (x86_64)
--- | ---
Revision | 5bc145d831c180d9ff94f29a0d7a2e1cbd30ef36-refs/branch-heads/4430@{#1311}
OS | macOS Version 11.2.3 (Build 20D91)

• verification notes can be found via https://github.com/brave/internal/issues/766#issuecomment-824977136.

---

Verification passed on

Brave	1.23.73 Chromium: 90.0.4430.85 (Official Build) (64-bit)
Revision	5bc145d831c180d9ff94f29a0d7a2e1cbd30ef36-refs/branch-heads/4430@{#1311}
OS	Ubuntu 18.04 LTS

• verification notes can be found via https://github.com/brave/internal/issues/766#issuecomment-824977136.

---

Brave | 1.23.73 Chromium: 90.0.4430.85 (Official Build) (64-bit)
-- | --
Revision | 5bc145d831c180d9ff94f29a0d7a2e1cbd30ef36-refs/branch-heads/4430@{#1311}
OS | Windows 10 OS Version 2009 (Build 19042.928)

• verification notes can be found via https://github.com/brave/internal/issues/766#issuecomment-824977136.