Re: Which modern LP Type?
Out of the ones I've played, if you ask me what a modern LP is I'd say it's a Jackson Dominion. I know it barely looks like a Lester, but to be honest a Saber is pretty far from a '54 Strat too and it's still considered a superstrat.
The basic recipe is there: Mahogany body, maple top, mahogany neck, 24.75" scale, '59 pickups (recently switched out for DiMarzio Dominions which are decidedly more modern but still not super high output).
The modern upgrades: oiled necks, chambered body, graphite reinforced necks, coil taps, an uglier but more ergonomic body (IMO), neck through, compound radius fretboard, and locking tuners in some models.
Edit: Also, if you're looking for a cheaper model, wait for the DX2 that was just announced. X series instead of Pro and maple neck instead of mahogany. Still comes with DiMarzio Dominions.