What is an appropriate measure to record your weight?

That depends on where you live.  The purpose of writing down your
weight is often to tell other people what your weight is, so you want to
record it in units that the other people around you understand.

If you live in rural Scotland or Ireland, then you'd want to record it
in terms of stones and pounds.

If you live in Liberia, Myanmar, or the USA, then you'd probably
weigh yourself and record the results in pounds.

Anywhere in the rest of the world, it would be recorded in terms of kilograms.

You must be especially careful in the USA.  Here, you must not allow
any metric units to appear anywhere in your records, for if the eye of
an American should chance to fall on any written mention of a liter,
a meter, or a kilogram, the effect on his American physiognomy is
toxic.  His blood evaporates, his eyes turn backwards in their sockets,
and his abdomen falls out onto the floor.  Worst of all is the effect on
his brain.  It is well established now that having been born in the USA,
or after living there for some period of time, a person is incapable of
comprehending the metric system of units, so the rest of us must
protect them from it.