I recently purchased a pair of Lake shoes in size 42...which according to their website would be a US size 9.5. "https://lakecycling.com/pages/sizing-charts"

The box that my shoes came in claim a 42 is a US size 8...note the photos below of other shoes posted for sale

The yellow shoes are a Euro 43.5 and a US size 9.5

The sticker on the next box below claims that a 43 is a US size 10

So according to Lake: 42 = 8, 42 = 9.5, 43 = 10, 43.5 = 9.5

Does anyone have any assistance as to this rationale?

Do they fit?

Size designation doesn't really matter - it's going to vary from manufacturer to manufacturer and model to model. Rather than kvetch about the inconsistent sizing conversions on the aforementioned chart, you should be thankful for their length and width specs for each last, each last width and each size on that very same chart - very, very few manufacturers go to such lengths.

There is no rational.
You have to measure your foot, preferably in your cycling socks and compare it to the chart.
I know that for the same size foot you would order a different EU size for SIDI as compared to Giro. Knowing how big your foot is = better than a shoe size.

You can find a sizing chart for all the major shoe brands with the internet!
SIDI
Giro
Specialized
Shimano