The height profile shown there suggests Milton Keynes has gained about 200m during the day. I guess this is because height is measured by atmospheric pressure?
Is there any other more accurate way of measuring height climbed on a ride?
It's more like 130-140m than 200m (compare distance between 100m and 250m labels with the height at the right).
Normal pressure changes during a day would be more like 40-50m, and you'd expect to have noticeable windiness once it got to 150m during the day. At the extremes the difference between a good high pressure, and the middle of the eye of a strong hurricane would be around 3000 ft/900m.
GPS Barometric altimeters are accurate to around 1m (i.e. difference in height between floor and overhead will show at 2m, subject to rounding errors). The problems are height drift due to pressure changes between calibration and reading, and you can get small dynamic effects due to air moving around the unit (eg when I ride into a headwind, air speed maybe 25 mph, my eTrex reads 3-4m higher if I hold a hand in front than if I don't).
GPS heights are very good if you average them for long enough in the same place, but any one height reading (at once a second) will be in error with a standard deviation of around 10m (about 3x horizontal error). The question then is how long to average the heights over? Too short a period, and the random fluctuations will add into the total climbing, too long and you'll miss out on short-lived local extremes. Garmin don't tell what averaging they do, but GPS climbing is normally significantly higher (like 15%, ish) than either barometric (on a good day) or contour counting.
Garmins that have a barometer will generally calibrate it from averaged GPS height when first turned on, then use & log the barometric height. For most accurate figures, you ought to leave the GPS turned on and stationary for 10-15 minutes before you start to ride, so that real height changes don't confuse the calibration. I don't suppose most people would bother, but you could turn on first, before doing tyre checks, clothing adjustment and other faffing.