No, it doesn't. In the context of para 5 of the PDF it applies only to EASA aircraft.
Para 5 describes the EASA FCL Annex III "Validation" of 3rd country licences. Validations can be used to fly any aircraft requiring compliance with EASA FCL - ie. a UK one, as in the example in para 5, any EU registered one as you say, but also any FRA operated by an EU resident who needs to comply with EASA FCL. At least that's how I read it.
Put another way, I think the (formal or informal?) definition of "EASA aircraft" (as opposed to "EASA-registered aircraft") includes 3rd country aircraft operated within the EU by an EU resident?
brgds
421C