After knowing that Europe developed energy saving T8 fluorescent tubes rated at 18w, 36w, 58w, and 70w to directly replace 20w, 40w, 65/80w, and 75/85w T12 fluorescent tubes on preheat ballasts, I am beginning to wonder whether those T8 tubes operate at lower currents than their T12 counterparts for example, would a F58T8 tube run at 115v 550mA compared to a F65T12 tube running at 110v 670mA?

They are supposed to replace the T12 on the same ballast, so to get lower power the only way to do is lower the arc voltage. And because of the ballast being not an ideal current source, it even means the current is even slightly (2..3%) higher. All that behavior is taken into account for the 18/36/58/70W specs. So if a 40W is rated for 430mA, the 36W formal rating is 440mA, because that are the currents delivered by the same reference (= with all parameters centered exactly at the nominal values, normally production has some spread around that) "40/36W" ballast into either one reference lamp.

What differs significantly is the (hot electrode) ignition voltage, but that was not what the question was about.

I have seen some people claim that F36T8 tubes run at 350mA at 110v. I wonder if that is an incorrect claim? In addition, I also see that whenever arc voltage gets lowered, the ballast is more likely to overheat. In North America, the F34T12 fluorescent tubes meant to replace F40T12 tubes save energy by lowering arc voltage by a large margin and that gives F34T12 fluorescent tubes their notoriety for overheating F40T12 preheat and rapid start ballasts. According to silverliner, he claims that F36T8 tubes do not overheat North American F40T12 preheat ballasts but yet reduce total system wattage. That leads me to think that European energy saver T8 tubes do not save energy by having the arc voltage reduced, but rather the current is reduced.

Do some math:
If you take the ballast load curve drawn as V=f(I) with V on X axis and current on Y axis (assuming just a resistive load and the ballast being perfectly linear and lossless reactive device, so no saturation,...), you get a quarter of an ellipse.
Now if you trace along the curve and express the power, you get a curve starting from zero (zero current, certain voltage), then as the current increases the power rises, but at some point the voltage starts to fall so down, the higher current leads to lower power because the voltage gets reduced way more.
If you want to calculate the point of maximum power, with a reactive ballast impedance you get it exactly at Vload = OCV/sqrt(2) and Iload = Ishort / sqrt(2).

For a 230V that maximum power point means load voltage of about 160V and current of around 0.36A so power around 57W.
So when the nominal 110V leads to 40W load, to get 36W you would need to either rise the voltage to about 200V so the load current will drop so to reach the 36W, or lower it to the roughly 105V. Because the arc voltage can not be higher than half of the mains, the only working option remains to use lower arc voltage.
Yes, technically the current increases, but we are talking about just 2% or so, way below the manufacturing tolerances, that can not have any influence on the real life reliability or so.

The reason why F34 appear to kill F40 ballasts is to me:
- mainly the coincidence: Many places switched to F34 quite late and all at once, so when the ballast failure happened, the F34 was in it. Because the F34 came at the time when the ballasts switched from PCB oil capacitors to PCB free materials, which initially exhibited shorter life. So the fact why ballasts have been failing sooner may have been caused by the ballasts being bad quality itself, but the F34 were blamed because it happened at the same time.
- The ballasts were already nearly shot, just virtually "hanging on a last thread". Then even a tiny bit extra stress from the lower arc voltage could be enough to break that "last thread".
- The F34 may exhibit greater ill effects when it started to fail, namely more stable arc, so allowing way stronger rectification effect (when one cathode is dying sooner, the F40 just completely quits sooner, but the F34 holds its arc longer, when the rectification is stronger). This rectification effect then causes either higher DC current component (lag sections) or a higher peak voltage across the series capacitor, causing it to fail sooner.

Which of the three  is the strongest, I don't know. Likely each of them is a contributing factor to the bad F34 reputation.

Does this mean that a F36T8 tube is severely underdriven if it is running at 350mA and that a F58T8 tube is severely underdriven if it is running at 550mA?

Does this mean that a F36T8 tube is severely underdriven if it is running at 350mA and that a F58T8 tube is severely underdriven if it is running at 550mA?

"Severely" may be a too strong word (it is about 20% lower than rating, that is within the range still tolerated quite well, although the light output will be off), but yes, they are underdriven. Assume low frequency AC drive. HF drive usually leads to about 10% lower current to reach the same power, plus another 10% sometimes counted on the higher efficacy at HF compare to mains frequency, so with HF drive it would lead to about the rated light output, so being the correct current. All currents are assumed to be of near sinewave shape and the value represents the rms value.

"Severely" may be a too strong word (it is about 20% lower than rating, that is within the range still tolerated quite well, although the light output will be off), but yes, they are underdriven. Assume low frequency AC drive. HF drive usually leads to about 10% lower current to reach the same power, plus another 10% sometimes counted on the higher efficacy at HF compare to mains frequency, so with HF drive it would lead to about the rated light output, so being the correct current. All currents are assumed to be of near sinewave shape and the value represents the rms value.

I have seen some members say that F36T8 tubes actually run at 350mA when used on their intended ballasts to achieve energy savings rather than running at 430mA for F40T12 tubes on the same ballasts.

The 34w lamp is rated at 460mA and 72v arc volts. I’m not an expert on the operating specs of the European T8s but I find 1940s preheat ballasts to run cooler with these than 40w T12s. 34w T12s causes the same ballasts to run warmer.

I have seen some members say that F36T8 tubes actually run at 350mA when used on their intended ballasts to achieve energy savings rather than running at 430mA for F40T12 tubes on the same ballasts.

That could be when you want to achieve with a T8 just the same output as there was with the T12. But that is not the official rating, nor power, nor lumen output.
An in EU ballasts driving lamps so they don't reach full rated output (with dimmable ballasts it means at full brightness setting) are illegal already for at least 40 years (first big group of the energy efficiency regulations in Europe come from the 70's energy crisis, like ballast efficiency effectively banning all resistive fluorescent ballasts, requirement to drive the lamps at full output,...), maybe even way longer, so such fixture was for sure not sold here, at least legally. Or to be more precise, fixtures or lighting equipment is not allowed to drive lamps so they don't reach their full output. It also includes those shady "incandescent life extenders", which prolong the lamp life just by underdriving them so cause them run in a very inefficient way.
So ballasts you speak abut culd be sold and installed somewhere overseas, where they use "EU" style 220..240V mains so where the "European spec" lamps are marketed too, but do not have the efficiency regulations in place.

Don’t forget that a 36W tube will have 32W on electronic ballast! Same for the 58W with you get 50W on electronic ballast.

These are data from all major manufacturers.