how do the two speeds inter-relate?
They don't. LAN speed and internet connection speed are not related in any way. Neither are wifi speed and wired speed on the same router / access point.
why such a high LAN speed
Basically because the technology came from corporate computing to home computing.
The original Ethernet specification was for 3Mbps, shared between all nodes on a segment. This became 10Mbps in the early 80's and remained at that until the mid-90's, when 10base2 and 10base5 thin and thicknet bus networks were superseded by the now-familiar star topology running 10baseT (RJ45 twisted pair), and an advance in ASICS and cabling standards allowed 100Mbps to be introduced.
Initially aimed at backbone and aggregated connections, it quickly became cheap enough to implement on all network cards and switch ports, so the de-facto ethernet speed standard became 100Mbps (with a backwards compatibility with 10Mbps) - and that spread equally to the home network, as these became ubiquitous with the take-up of the internet.
Now it is common to have network cards supporting 1000/100/10 Mbps, simply because it is as easy for the manfrs. to do so than not.
The same is not yet true of switchports, and your bog-standard switch will support 10/100Mbps, even though the upstream connection to the ISP may be less.
A LAN speed (hard-wired) should show either 10 or 100Mbps; anything else is an error, possibly due to incorrect autosensing or duplex mismatch.
A wifi LAN speed could show anything from 1 to 150Mbps, depending on implementation and signal strength.
SD