First question would be how to define a democracy and in what form. If you define it in modern sense then Romans might be the first ones, but even that is debatable. We should instead look at how ancient authors would define it. Aristotle would define it as: "It ought not to be assumed, as some people are nowadays in the habit of doing, that democracy can be defined without any qualification, as a form of constitution in which greater numbers are sovereign." (Politics 1290^a 30) So biggest Ancient source and theoretician of democracy would not just define it by the width of the electorate. His criterion was numbers, poverty (if rich vote then it is oligarchy) "There is a democracy when the free-born and poor control the government, being at the same time the majority; ..." (Politics 1290^b 7).

What you defined as democracy, ancient authors such as Aristotle would define as politea or as we might call it, a republic.

Now I want to address you question from the title, whether Rome was a first democracy, if we decide to define it as such. That is a good question, both Athenian and Roman political systems started to develop around the same time with Athens having a head start with Solon of some 50 years to Rome. By even ancient standards we might give the title of first democracy to Athens but even that is questionable.

Herodotus in his History noted this: "When Mardonius arrived at Ionia in his voyage by the coast of Asia, he did a thing which I here set down for the wonder of those Greeks who will not believe Otanes to have declared his opinion among the Seven that democracy was best for Persia. Mardonius deposed all the Ionian despots and set up democracies in their cities." (Historia 6.43)

What we can define is the fact that political systems we call democracies started roughly around the same time with different reasons why.

Roman republic mostly developed it's system as a kind of a response on the rule of Tarquinus Superbus and as such was designed not to enable another autocrat to come into power. Of course now we know it failed but the reason why it held up so long was the fact why and how it was designed.

If you read Roman laws you can see that personal liberty of every individual was extremely important to Roman lawmakers, something that was under threat during the rule of Tarqiunius Superbus.

Sources:

Aristotle. Politics. Oxford University Press, 2009.

Herodotus. The Histories

Skinner, Quentin. Liberty before liberalism. Cambridge University Press 2012.