In article , Nicolaas Vroom
writes:

This is what most people refer to as "the observable universe" or "the
part of the universe within the particle horizon".

The question is if this definition is independent of humans. IMO it is not.
IMO the radius R (R=1) of the friedmann equation is independent of humans.
It should describe all that is changed (created) as the result of the
Big Bang.
When you study Supernovae 1A the distance is a function of "humans" i.e.
it is a function of the position from were the distance is measured.

Right. The observable universe obviously depends on our position: we
are at the centre of it. It is like the horizon on Earth: the average
person can see about 11 km if there are no obstructions.

When you study the distance from which light can reach you it is not that
simple.
Locally at present this distance is zero. When you go backwards in time the
distance increases lineair with time in the past. However the further
away you go backwards in time the increase diminishes because space expansion
has to be taken into account. When you go back more (roughly 7 b years after
the BB) the distance is constant. More earlier the distance from which light
can reach you decreases. At the time of the Big Bang the distance is
again zero.

What you are referring to is the proper distance at the time the light
was emitted. In the context of the obervable universe, we are
interested in the proper distance NOW of the most distanct objects which
we can see.

When you consider the maximum distance (as defined above), then this
distances
increases lineair as a function of the time after the Big Bang.
However the space occupied is much smaller as Our Universe.
Inflation has "nothing" to do with this.

True. What Tegmark means is that if the universe is almost flat as a
result of inflation, then the universe (his Level I multiverse) is much
larger than our observable universe (his universe). This is true
without inflation as well, of course. However, a nearly flat universe
is a robust prediction of inflation (non-flat inflationary models are
contrived), so his statement is true: from inflation follows that the
universe is much bigger than the observable universe. It doesn't
REQUIRE inflation, though.

Your description of the distance at time of light emission holds for all
geometries. If the universe is positively curved with a radius of
curvature just a few times the Hubble length, then there is more to the
universe than we can see, but not THAT much more.

By symmetry, it is the sphere defined by photons which left our
position in space at the big bang. It always increases in size
in co-moving coordinates.
We have to be carefull. Our Universe increases in size as described
by the friedmann equation.
How ever that is not what we observe.
At present of our universe we can only observe a small part.
Out side that space we can not observe galaxies in their present state.
However those same galaxies we can presently observe when they
were younger and the distance towards us was smaller.

Still, the radius of the observable universe always increases in
comoving coordinates. The observable universe is, by definition, that
part of the universe which we can in principle observe.

IMO that makes only sense when we study supernovae 1A. See above.
Of course when we study the evolution of "Our Universe" it should
not be in conflict with what is observed.

There are more things in the universe than just supernovae Ia. :-)