ETA I note that the first report you cite, at "iceagenow.com", says about the author:
Felix is not affiliated with any university, scientific establishment, or corporation, and therein lies his strength. Untainted by institutional bias or conventional wisdom this architect turned author brings fresh insight to the study of the ice ages.
I suggest that he is probably less reliable than scientific consensus; peer review has its flaws but does exist for a reason! If you would like a close reading and critique of his report, can you make that clear? Cheers :-)
The authors attributed these effects to decompression melting. If you remove a weight from above a magma chamber, the pressure in the chamber is reduced, and the melting temperature is lowered, causing an increase in the amount of melt available for eruption. (Personally, I suspect that it's not really decompression melting causing the change in eruption rate but actually just fractures opening and closing.)
The Tolstoy paper claims that if you look at seafloor topography, you see bumps where more lava erupted and piled up higher. Further, it says those bumps correlate with the 100,000 year Milankovitch global climate cycle. So, orbital eccentricity --> sea level --> midocean ridge eruption rate --> seafloor topography. This is a brand new idea and needs much more testing. (Personally, I find the topography thing hard to believe because that topography is probably more fault-controlled than eruption-rate-controlled.)
Tolstoy even claims that low tide causes eruptions! She uses a Schuster test to say that the majority of seafloor eruptions occur during neap tide. However, we don't observe seafloor eruptions very often, so she is making this claim using only 9 data points.
It's currently unknown whether these unloading / loading effects are important at subduction zones or LIP.
You've asked a difficult question which volcanologists have been attempting to understand for decades and will continue to work on for the foreseeable future. In summary, my answer is that cutting-edge research suggests that global eruption rate fluctuates in response to climate via changes in the masses pushing down on magma reservoirs.