I am sorry I did not make clear I offered that example only in response
to your comment about the meaning of "pay". (I'm not aware of a
definition of "sexual services" as such but thought the definition of
"sexual" in s.78. was rather important for the interpretation of the SOA
2003)
As for ss.(3) not defining "payment", I don't know what else one would
call it. In the context of statutory construction a provision that "X
means" is conventionally referred to as a definition of X. And my point
was that in that section "payment" was *not* just being left to take its
ordinary meaning from ordinary usage and earlier cases on the meaning of
"pay".
There is a related issue, much debated, about the pros and cons of
provisions which (explicitly or implicitly) "deem" ordinary words (such
as "payment") to have a different (usually but not always) wider meaning
for the purpose of some legislation. On the one hand it makes for
shorter and simpler legislation than coining a specific term (eg "sexual
services reward") which most readers find easier to follow. On the
other hand it risks readers taking it at face value without bothering to
read on to find that payment includes, as you recognise, things which
would (very probably) not count as payment.
But s.51 is a bad example to use for all that as it defines "payment"
only for the purposes of ss.(2). That's now deprecated in Whitehall so
it might be that if the Act were written today the definition of
"payment" there are in other sections might be rethought.