For the USA or UK, the German example is probably easiest to follow.
After WWII, Germany took over the US voting system. They have states, and they vote in districts. Because winner-take-all is by default not a true democracy (the losers do not receive representation of their choice), the Germans fixed that problem.
Let's take California's Senate as the example with 40 districts. Allow me to say this simplistically (in reality German voters vote twice in one election).
Applying the German system means first of all that gerrymandering becomes unimportant toward the outcome. That is an immediate win. Voters vote whomever they want to vote for and the plurality vote wins the district seat. That delivers 40 senators.
Next, all votes throughout California are added and it shows, for instance, that the Green Party got ten percent of the overall votes.
This result is then translated into the outcome. If the Green Party did not win any district, then 4 senator seats are added, making the total 44. In reality, voters in one district may have given the Green Party one seat. In that case, 3 seats are added, total 43 (but Green Party then still having about ten percent of the seats).
This continues the tradition for voters to have a district representative, while all voters did end up being represented in the place where the decisions are made. That would make the German voting system a democracy.
In Germany, however, a 5 percent threshold was put in place at the overall level. Only parties with more than 5 percent of the votes would get seats added to the whole of district seats. In case a small party wins a district, then they would still get that seat, even when not reaching the 5 percent threshold overall. So, the German democracy is undermined in regards of not respecting all voters. The splinter parties do not receive their natural seats. Still, Germany reaches a reasonably high level of political freedom.
Prior to this system, Germany had equal representation, which is truly wonderful for small nations. Interbellum Germany, however, showed us that the most accurate form of democracy can behave in undesired ways when there is a crisis, particularly important for large(r) nations.
In the early 1930s, the communists won about one-third of the votes, and the fascists won about one-third of the votes. The parties in the middle were basically splinter parties (to keep the story simple) and the pressure to cave to one side was enormous due to the global crisis that hit Germany especially hard. The splinter parties had Stalin and Mussolini as examples and they ended up handing their Austrian Mussolini the scepter.
As a side note, it is interesting that one and the same crisis caused voters to split toward extreme right and extreme left. Both political extremes proffered solutions, both lying like snakes. The US ends up being the hero, because the real problem, UK and France imperialism (good for them, bad for the others) is called out by the US as the culprit. US sides with colonies becoming independent.
In general, modern Germany has had between four to six parties with the new and revised US system, delivering both decent stability and decent political freedom.
New Zealand changed its two-party system in 1996, basically using the German system. Female representation jumped almost 50 percent overnight.
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The US Constitution is an interesting document in that it declares only how the Federal level needs to function, election-wise, such as with 2 senators per state.
The US Constitution does not declare how elections on other levels of government need to be held. In general, all governmental levels mimic the Federal form of elections, but nothing tells them to do so.
The 14th Amendment has been ruled to demand of governments that they do not use discriminating systems serving the public if a better system is available.
This does not apply to what the US Constitution itself declared (2 senators per state discriminates voters in California, for instance, because they get only 2 senators whereas voters in New Hampshire get the same number of 2 senators as well, which is pure discrimination from a voter perspective).
The States are also given freedoms to do things as they please per the US Constitution.
Cities and counties are not mentioned in the US Constitution at all, so there are no exclusions or freedoms provided for them. They must follow the US Constitution and cannot have discriminating voting systems in place. The States are not given the freedom to hand out US Constitutional freedoms to cities or counties. Today, cities and counties operate elections in violation of the US Constitution, per the 14th Amendment.
That means that change can be put in place reasonably quickly, but with the system now in place this will in general require public support (i.e. education about voting systems needs to be made available; now no one is educated about voting systems and their effects). My prediction is that voter equality is going to be the ultimate civil rights movement that will finally emancipate the United States.
For the States, there is the freedom to change their system today, so it can get done without going to court. However, there is no Constitutional demand. Once parties (political groupings) appear at the local level, parties will end up getting voted in at State level, too. This process will educate the public even further about voting systems and their effects on the decision making process.
I did the calculations for the US Senate in 2006 (to see how the system discriminates), and just shy of 60 percent of the voters could point their fingers toward the individual they voted for.
Having A/ majority-win and next B/ majority-rule translates with this 2006 example into 30 percent of the voters effectively having picked the representatives that made all the decisions. Another way to see that the US is not a democracy according to the definition of what a democracy is.
[The Republicans had the majority of seats but this was based on fewer votes than the Democrats had received, showing how the system also discriminated because of the 2 senator per state rule that benefited the Republican Party].
In pure democracy, the majority decision is directly based on the majority of voters. In all other forms of voting, the majority decision is based on a lesser number of voters (and can basically always be minority decisions).
