First off, AMD64 = x86-64; they both refer to the same architecture, not sure why they used a different nomenclature.
Secondly, Oracle Database 10g Instant Client for Linux AMD64 (32-bit and 64-bit) comes with two different packages available for download, while Instant Client for Linux x86-64 comes with a single package available for download.
This is because Oracle Database 10g Instant Client for Linux AMD64 (32-bit and 64-bit) comes with a different package for each architecture (one for x86 and one for AMD64 / x86-64), in fact the size of the two packages amounts roughly to the size of the package in Instant Client for Linux x86-64.
Since you're running a 64-bit version of Ubuntu, just pick the 64-bit version in Oracle Database 10g Instant Client for Linux AMD64 (32-bit and 64-bit), which is only ~32MB vs the ~62/~63MB of the multi-architecture version in Instant Client for Linux x86-64.