Interpretation could be defined by how the performer applies expression to a piece.
A performer could play a piece without expression (e.g. Glenn Gould's Bach) by playing it in strict time with little consideration to the dynamics and he can be said to have played it dry, lifeless, emotionless, etc. This is what some people call "interpretation" as he interpreted Bach's compositions in a boring and dry manner.
Contrary to Gould, Tureck plays Bach applying all of the expressive techniques and it is very much in contrast. Her "interpretation" is that Bach's works are full of emotion and should be expressed as such.
Some performers apply expressive techniques even though a certain phrase cannot be expressed in such a manner. It could be they alter time (tempo rubato) or the dynamics when the phrase does not call for it. Exempli gratis: most performers [I have heard] of Beethoven's sonatas apply expressive techniques when it is not called for. Why do they do this? Perhaps due to lack of control, musical taste, etc. (E.g. Schnabel - tempo is irratic, dynamics too sudden; many others suffer from this syndrome). This is what some would call "over interpretation" - applying expressive techniques when they are not called for in a piece.
Why not call "interpretation" as "expression"?
There are two aspects to musical expression: it is the control of...
[0. playing the corrects notes is assumed]
1. time
2. dynamics
(3. tone - unique to each individual instrument. With the piano, it is the use of the pedals, etc. But to keep it simple, I ignore this aspect.)
Time is the tempo that a piece is played and how it changes within a piece.
Dynamics is the loudness a piece is played and how it changes within a piece.
These two are not seperate entities - they are directly related.
Expression is how a performer varies these two aspects to create a desirable musical effect within the context of a whole piece of work. Some apply it too little or not at all while others apply it too much and over the top. Both are incorrect.
A slight digression as I have refered to expression as an entity outside of music but must be mentioned: expression is not something that is to be added afterwards, it is there from the very beginning.
Interpretation is not the reading of the expressive markings refering to dynamics or time. These include: p, f, cresc., dim., allegro, adagio, dolce, appassionato, etc. Interpretation is the reading of the notes and what they convey. The expressive markings are there to aide in the interpretation of the notes which leads to "expression". In other words, expression is a syndrome of interpretation. Without the ability to interpret the notes, time, and dynamics, one would not be able to express a piece.
And now on to a digression majora:
The use of words to describe music will always be inaccurate because they are two different entities. Historically, there is the poor use of words to desbribe music; music is almost always described using less-than-concrete terms like adagio, andante, allegro, presto, dolce, fuoco, agitato, forte, piano, etc. To understand what these words refer to require extensive experience with them, not just a theoretical understanding. Due to the ambiguity of words, composers have become ever more conscious of what they write in their compositions. Bach didn't write any in his music. Beethoven wrote many and Debussy went even further. Now we have specific metronome markings to refer to the tempo.
How do we play something dolce? Legato? Leggierra? Agitato? Appasionato? We can't unless we have the notes, dynamics, and time. Can we play the beggining of the third movement of the Moonlight sonata dolce? No. Why? Because the collection of notes and time do not allow for it. Can we play it agitato? Yes! Why? Because the collection of notes and time allows for it. Can it be played appassionato? Yes... but then no. Why? Because the first few bars repeats but slightly differently and the surge of the crescendo to the sf loses its effect if played another time. But can the first bars up until the sforzando be played appassionato? Yes, but only if taken out of musical context (the rest of the movement is not played).
And a super majora digression would be on how to concretely define expressive markings.
Best intellectual fervor,
fDsF.