How do the 'EFI system partition' and UEFI *.efi files relate?

No, a partition is not one and the same as a file; and Wikipedia is misleading you.

That second paragraph is wrong on pretty much all points. EFI does not require any such things, and never relies upon a boot sector. If you want the gen on this, I suggest reading the actual EFI specification. It's fairly clear on what the \EFI\BOOT\BOOTxxx.EFI files are all about. They are the fallback default if nothing is configured in the Boot Manager. Normally, however, something will be configured in the Boot Manager.

EFI does not require a special partition table, it being capable of using both the old PC/AT (a.k.a. MBR or MS-DOS) partitioning scheme and the new EFI partitioning scheme. Nor are EFI boot applications specially located. Their locations are stored as paths in individual boot options. In theory, they don't even have to be on the EFI System Partition. They could in theory be on any volume whose filesystem format the firmware understands. In practice, the ESP is both the expected place that users will look at first and is guaranteed to exist by the platform and be locatable by EFI programs.

Further reading

• Jonathan de Boyne Pollard (2006,2010,2011). What "boot" and "system" volumes are. Frequently Given Answers.
• Jonathan de Boyne Pollard (2011). How operating systems determine the location of the system volume when bootstrapped. Frequently Given Answers.
• Jonathan de Boyne Pollard (2006,2011). The EFI boot process. Frequently Given Answers.
• Jonathan de Boyne Pollard (2011). The PC/AT boot process. Frequently Given Answers.