The top two competing ideologies in the Democratic party seem to be Neoliberal and Progressive. Which one fits best?

Those definitions don't seem to be mutually exclusive, or even significantly in conflict. I think finding other, more precise definitions, which better delineate the conflicts, would lead to a more helpful answer.
There are some significant differences. For example Neoliberalism requires free market capitalism while progressivism does not.
There are two competing American ideologies: baseball fans and vegetarians.
I don't see that divide. I see the divide between reactive "panacea progressives" and actual progressives.
Panacea progressives was new to me I had to look it up.
Okay, but what about reactive. That part doesn't make sense. In fact Conservatives are the reactionaries. I'm not able to follow this new grouping. Who would you call a reactive panacea progressive?
Which one fits what best?
I'm also confused about what question OP is trying to ask.
The Democratic Party.
I’d disagree. Social democrats vs neoliberals.
It’s an alliance of soc dem and socs to be more accurate. But 80% of the policy is just soc dem so probably fair.
I see it as champagne socialists versus apathetic centrists versus technocratic progressives.
What significant differences do you see between Social Democrat and Progressives?
The two things aren’t mutually exclusive.
Neoliberalism is just an internationalist market orientation. Progressivism is social engineering. Most nations today have both.
Are these personal definitions or textbook. I've never heard of that before.
How would you define internationalism? This is sort of a side discussion, but I would be interested to know.
To me, internationalism runs completely counter to neoliberalism, in fact they are contradictory terms.
Internationalism is the idea that workers should organise internationally. This idea is expressed by Marx in the Communist Manifesto, when he writes: "Workers of the world, unite!". According to Marx, class was the ultimate divide, not nationality.
Neoliberalism, on top of being hostile to worker organisation, can also manifest itself as openly nationalistic. If we examine neoliberal figures, such as Pinochet, Reagan and Thatcher, they can definitely be described as nationalists, or at least conservatives.
On to the main issue I have with your comment.
How is progressivism social engineering? What do you mean by that? I assume it is a criticism of progressivism, or maybe just an assertion that it only has to with social attitudes.
Progressivism denotes a political philosophy that is based on the basic idea of progress in different areas of society, from science to economics.
The Progressive Era of your country was marked by progressive social action tackling the problems caused by industrialisation, urbanisation and corruption.
Unionisation and the introduction of anti-trust laws and income taxes were a large part of this. Neoliberals are not fans of unions and Reagan's administration weakened the Sherman Antitrust Act, so these the policies of progressivism are not just not counter to neoliberalism, but the neoliberal era marked the end, or at least the weakening, of these progressive measures.
Progressivism is therefore not just contrary to the social conservatism of Pinochet, Reagan and Thatcher, but also contrary to their economics of neoliberalism.
Neoliberalism and progressivism cannot be reconciled. They are clearly two fundamentally differing worldviews. To answer the question of this thread, the Democratic Party should embrace progressivism and leave neoliberalism to the Republican Party.
I feel like you misdefined both. I think a better question would be there are two competing ideologies, incrementalism vs accelerationism which one is better. I think most neoliberals support science and I think most anti-science people on the left (Vaccines made my kid autistic GMOs are killing us etc) are anti-capitalist.
Do you have another mainstream definition to offer? Because right now it looks like you're dismissing accepted definitions in favor of your opinion.
Neoliberalism has a home in the Republican Party, not the Democratic Party.
The word is just liberal. Free market capitalism is no-qualifier-needed liberalism. Come on, no serious politician has ever called themselves neoliberal.
The argument between the Democratic Party right now is whether we’re going to be the anti-wealthy party or not. Being pro-science is something they tell themselves as they try to nationalize pharma and being down Elon Musk for being anti-worker. You tell these guys that technological advancement comes from corporations and they blow their heads off.
I don't think Richard Spencer calls himself a racist either.
No one unironically uses the term "neoliberal" in electoral politics because it's basically a slur in the modern-day. We can still describe someone as one if they fit the definition of the word.
Of course, the term neoliberal is also super vague and not particularly useful in most conversations for that reason, but OP did provide a definition of the word above.
The divide is whether the party should focus on opposing the rich or focus on helping the poor.
I'm not familiar with this claim. Are you suggesting that free market capitalism is a necessary part of liberalism?
Only some technological advancement comes from corporations, and it tends to be stuff towards the end of the technology readiness cycle. The high impact stuff is more fundamental research, where government institutions carry a very heavy share of the load. NASA is estimated to have a 40:1 ROI for our economy. Don't even get me started on what NIH, DoE, DARPA, etc have done. All of these and more government institutions have had a contingent role in creating the technological world we have today.
Silicon valley itself owes its origins to partnerships between the government and educational institutions like stanford during WW2.
The word liberal is parent category for both progressivism and Conservativism. Neo liberals like to identify first as liberals because they don’t support progressive policy but, identifying as liberal allows them to create vague false brand.