For purposes of this comment, I will be attempting to clarify differences between social democracy, democratic socialism, and socialism-in-general.
>What exactly is different between the ideology underpinning Democrats, and socialism?
It comes as news to me that the Democrats have an ideology. Ever since Reagan, and then Clinton, their standard strategy has been every bit neoliberal, even if not sadistically neoliberal like the Republicans have been.
That is, the difference between Democrat and Republican is, in my opinion, the difference between saying, "We're so sorry you're homeless, but if we tried to give you housing, businesses would flee the country, but at least we're using government funding to help keep your soup kitchen open", and kicking the homeless while yelling "Get a job, you filthy bum!".
Notice that in both cases, the homeless person remains homeless.
Well, I suppose if we can call pragmatic-to-the-point-of-fundamentally-unprincipled meliorativist humanitarianism an ideology, then they do have an ideology, but only the very leftmost portions of the Democratic Party actually begin to be social democrats at this point. Elizabeth Warren, for instance, is an actual social democrat: she favors a broad ideology of decommodification, in which the necessities of human life become social rights and a middle-class society is explicitly encouraged via policy. Actually, she only even begins to favor decommodification.
>The Democrats obviously support socialized medicine, socialized education, wealth redistribution via high taxes, and significant regulation of private enterprise.
As noted above, in terms of their voting record, they mostly don't favor these things at all.
Also, "significant regulation of private enterprise", in terms of things like OSHA, the EPA, occupational licensing, Sarbanes-Oxley or Glass-Steagal, etc, should be placed completely to the side. Everyone who's not wildly insane is in favor of significant regulation of private enterprise, and the only negotiation is over where the boundaries of pragmatic, useful regulation actually fall.
(That is, most people who are not right-proprietarians (ie: not in the Libertarian Party) favor regulation of business for pragmatic, instrumental reasons, rather than considering it always terminally valuable as a point of ideology. "Reversed right-proprietarianism is not leftism" is a slogan you should repeatedly write on a chalkboard until you manage to actually believe it.)
>For instance, maybe a "true socialist" would favor the state owning enterprises, rather than massive tax-and-regulate, but to me, that is not an important difference.
A true socialist favors workers owning enterprises. Whether this should be achieved via labor actions (strikes, etc), seizure of state power by revolutionary force, state power won in elections, or cooperative entrepreneurship is a matter of strategy among socialists. Sometimes we disagree with each-other as points of ideology, and sometimes as matters of pragmatic implementation. Sometimes we agree, despite hailing from very different schools of socialism.
As a classification heuristic, "revolutionary" socialists favor seizing state power via revolution, "democratic" socialists (eg: Bernie Sanders) favor obtaining state power via electoral participation, and revolutionary anarchists favor using a revolution to destroy state power (never to be wielded again). All of these groups often adopt each-other's tactics if deemed pragmatically useful, so, for instance, almost all socialists support workers' cooperatives, even though many don't consider them the final goal of socialism.
Once again: reversed right-proprietarianism is not leftism, so I thank you for actually taking the trouble to ask a leftist what we actually believe. To repeat and sum-up the core point: socialism does not really care about taxation or regulation except as tools, it cares about workers owning their own tools, work-sites, materials, and enterprises, and thus controlling their own lives rather than being controlled by an owner.
>Hence, as a rule of thumb, I consider all Democrats to be socialists.
Basically nobody who calls themselves a socialist considers the Democratic Party to be socialists. Even the Democratic Socialists of America - who are often accused by other socialists of shilling for the Democrats, and who, truth be told, straddle the line between actual democratic socialism and social democracy - do not consider the Democratic Party to be a socialist party.
>So, I don't think Hillary or Obama is much different from Sanders, except that he is more bold and open and labels himself differently.
Clinton (both Clintons) and Obama have never put forward serious programs of decommodification (Sanders' social-democratic platform) or worker ownership of enterprises and means-of-production (Sanders' encouragement of ESOPs and cooperatives). In fact, Clinton and Obama have, by and large, not done anything for the working classes, instead preferring to appeal on a combination of identity issues and middle-class meritocracy-through-education. By their voting records, and in contrast to Sanders, both Hillary and Obama have voted in favor of treaties, regulations, and other forms of laws that not only took economic power away from workers and communities, but even away from industrialists, putting the nation's destiny in the hands of the financial sector.
From a left-wing perspective, Hillary and Obama are reasonably similar, while Warren and Sanders form a platform unto themselves, and Sanders borders actual democratic socialism (his platform is mostly social-democratic).