You have hit upon one of the most complex areas of the tax system. You would be well advised to seek advice from a CPA, enrolled agent, or tax attorney who is familiar with the US/UK tax treaty (i.e., deals with it frequently, not just once in a while).
That said, you seem to have identified two relevant sections of the treaty.
However, take a look at Article 1, especially the "savings clause" in paragraphs (4) and exceptions inj (5), and Article 24, especially paragraph (6). Pay careful attention to whether there is an exception in 1(5) that overrides the general rule in 1(4) that all income of a US citizen is taxable in the US, no matter what the treaty says. Your income may well be one of the exceptions in 1(5).
If you determine that the tax treaty overrides the US tax code and something that would otherwise be taxable in the US is not taxable in the US, then you need to figure out how to remove it from your US return and whether or not you need to file Form 8833 to disclose your treaty position. There are exceptions to filing the 8833. https://www.irs.gov/pub/irs-pdf/f8833.pdf (instructions after the form, "Exceptions from reporting"
If the savings clause means the US can tax your income, then you must include it on your 1040, but you may be able to get a credit for any US tax due on your 1040 via the foreign-tax credit (FTC) as required in Article 24.
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