Yeah, LBJ was a left-liberal not a moderate. Like Obama, JFK or FDR. You can count Billy and Truman as moderate liberals.
As already pointed out, most moderates are now in the Democratic Party since the GOP was sadly taken over by the religious right, science deniers, tea-party hacks and hardcore neocons. And now by nationalist right-wing populists and bigots alà Bannon. The GOP moderates remain at the state and local levels in more liberal states to be competitive. See Charlie Baker, Larry Logan or Arnold Schwarzenegger. They are definitely moderates.
Democratic Governors who are moderate:
John Bel Edwards, Jim Justice
Republican Governors who are moderate:
Charlie Baker, Brian Sandoval, Larry Hogan, John Kasich, Phil Scott, Chris Sununu, Bruce Rauner,, Chris Christie(?)
I would actually add Snyder to that list. He's definitely closer to the center than the State GOP.
