It's a team effort, actually. Checks and balances. Congress has the final say on what the laws are, though...
Sounds about right to my understandings, at least so long as the president doesn't ever get a line-item veto power. Congress passes bills, the president either signs the bills into law, or he vetoes them. Vetoed bills go back to congress, if they override the veto the bill becomes law, and as far as I know the bill becomes a law, and I'm assuming it does so without a signature.
The rest of your post sounds generally in accord with my historical understandings as well, I just wanted to make sure we were operating from similar background understandings.
Ike, to my perceptions was largely a non-partisan military man who favored some progressive ideas and preferences, but his essence was as a very moderate Republican (who frequently argued against the further right fringes of his day) and they were a large (if not the largest) part of the party through the '70s (with some exceptions, of course). The Dem party however, has almost always been made up of mostly neoliberals with a variable percentage of progressive influences and practices, but very few Progressive members or leaders.
my definitions of the terms I am using are probably in order here:
Liberalism - a political doctrine that takes protecting and enhancing the freedom of the individual to be the central problem of politics. Liberals typically believe that government is necessary to protect individuals from being harmed by others, but they also recognize that government itself can pose a threat to liberty.
Neoliberalism - a modified form of liberalism tending to favor free-market capitalism
Progressive - the application of rigorous analysis to reform current public policies to achieve more efficient and effective public policies addressing the needs and rights of the population.
Progressive Economics - understandings rooted in the concept of social justice which have the goal of improving the human condition often through government-based economic central planning. Economic progressivism is based on the idea that free markets are inherently unfair, favoring large corporations and the wealthy.
Both the Republican and Democratic parties in the US are neoliberal organizations, and both to varying extents and focus upon personal and social freedoms, pay homage to liberalism and free-market economics.
Progressivism is more of a process used to analyze and reform public policy while focused on providing efficient policies organized around the precepts of social justice and improving the human condition.