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All of southern England does - South West train services use a third rail as well, at least on most services - some of the ones to Basingstoke are diesel-powered, but definitely Devon, Dorset, Hampshire and Surrey have electric rails.

Ah, you got me. Not a true train buff :)

Does Essex count as Southeast England? Because the LTS (C2C) line is overhead-electrified. So is HS1.

More formally, I think they should have referred to trains that depart from London's southern termini? (Victoria, Waterloo, Cannon St, London Bridge, Charing Cross).