If the aircraft reduces vertical speed, surely then the aircraft is above the vertical profile? And you would gain speed by having to descend faster to get back on profile?
The VNAV profile takes into account the deceleration required, normally first at FL100 and then a second time approx. 20 nm from the runway. So if you let the aircraft fly in VNAV PTH (B737 term) you will see the aircraft decelerate without getting high on profile.
I have a vague understanding that being 'on profile' doesn't just include the altitude, but rather a combination of factors such as altitude and speed, and is therefore more about energy management than just height? Is that correct?
Sounds like your understanding is correct. If you are being held high by ATC you slow down towards Up-speed (B737 term again..) When you are cleared for further descent you can go down with a good rate of descent to while trading speed for altitude.
And if so, does that mean by slowing down as requested, even though you are above the profile 'height' you are actually still on profile because you are at a lower speed, and therefore have less energy to expend?
Slowing down usually means you will glide flatter, and that in itself will put you high. Increasing the speed will bring you down steeper and often required when provided shortcuts. Sometimes ATC (usually at smaller airports, especially those without radar) can tell you to slow down very early for separation purposes. So there you can be at FL300 at 220 kts with speedbrakes out to maintain profile (or energey management, if you will..) Not so pretty… but you do what you need to do.