As already mentioned in your question, in the Arduino tutorial "Melody" under "Hardware Required" is stated "piezo buzzer or a speaker" and in the comment that heads the sketch is stated: "8 ohm speaker on digital pin 8". But the pictures under "Circuit" and "Schematic" show a piezo sound transducer.
In any case the word "buzzer" is wrong. A buzzer usually has a built-in oscillating circuit or mechanism and, when energized, produces sound at a more or less fixed frequency. Better would have been: a piezo speaker and a 'normal' (moving coil) speaker.
There is a big difference between a piezo and a speaker: under a dc-voltage the piezeo wil have a near infinitive resistance and the 8 Ohm speaker roughly 4 Ohms. At 400 Hz the piezo has an impedance of roughly 2000 Ohm (there is a huge range of piezo's with varying characteristics), while the impedance of the normal speaker will be roughly 10 Ohm.
One side of the buzzer/speaker is connected to Ground, therefore current is only flowing when the side connected to pin 8 is HIGH (pin 8, has to source the current and not to sink it).
The datasheet of the Atmega328P states at page 299 that the absolute maximum ratings (which is not the same as maximum allowable operational ratings) are 40 mA for an I/O-pin is 40.0 mA and 200 mA for Vcc and Ground pins.
The datasheet shows a graph (fig. 35-24 at page 600) showing "I/O Pin Output Voltage vs. Source Current (VCC = 5 V)". From this graph an internal resistance of about 26 Ohm can be derived. The graph does'n show what happens at pin-currents larger than 20 mA, but let's guess the internal resistance stays about the same 26 Ohm.
When pin 8 is constantly HIGH (5 V at 0 Hz) the current through the speaker would be about 5/(26+4) = 167 mA. This is far above the 40 mA maximum rating for the pin current, but not enough to trip the USB-fuse (trips at 500 mA). When this situation lasts the pin-driver will be destroyed. This could cause an internal open circuit (with no further consequences) or an internal short circuit which might cause a supply current higher than 500 mA which would trip the USB-fuse. In this case pin 8 would be permanently destroyed which is easy to establish. If pin 8 keeps working, this scenario doesn't happen.
Normally the sketch "Melody" would not output a dc-signal (0 Hz). At 400 Hz the pin current would be 5/(26+8)= 147 mA. This is still far above the maximum rating for the pin current and the scenario is the same as above.
So the only way the USB-fuse could be tripped is when the pin driver burns out and causes a short-circuit from Vcc to ground. Reading between the lines, my impression is that Raddevus tried this out more than once, which means that pin 8 keeps working. Also I don't think that it is likely that an burn-out of a (few) transistor(s) of an IC causes a short-circuit. Therefore the cause for tripping the USB-fuse must lie elsewhere.
A last consideration is that Raddevus mentions (in his comment of June 18 at 15:34) three input buttons which are not in the 'schematic' shown with his question. He also seems to use a dodgy board ("missed ground hole on bboard, hit 5v"). So may be we don't see all the factors involved.