Yes, this is all true (although the concept of "citizen" is tricky). Spartan women were well known for being given considerably more autonomy, authority and respect than in Athens, which was a little Talibany. This is refereed to by many ancient writers, most famously Plutarch.

Edit: quick note that the knife bit may be a later romanticization by Plutarch, but everything in there is from the ancient sources.

de Ste Croix once said that “books on Sparta are numerous and mostly bad.” This is doubly so for Spartan women. The traditional view is pretty much like that tumblr post. You find this in works such as that of Sarah Pomeroy. Unfortunately I think a lot of it is fantasy. It stems from a failure to read sources in their context and a desire to find strong female role models in antiquity.

The problem with saying anything regarding Spartan women are the sources. Too many people elevate the ancient sources to the status of impartial and informed observers. Too few ask questions about the motivation, biases and pre-existing ideas which lie behind the creation of these sources. It is often forgotten that many sources originate in Athens and Athens was at war with Sparta at this time. It must be remembered that a strong empowered woman means a very different thing to us as it would have to an Athenian.

So lets look at some depictions:

To start with the Orestia. This was first performed during the first year of the Peloponnese war and the Eumenides includes a call for a permanent alliance with Argos- which would mean permanent war with Sparta. This is also the first time that Helen is made into the archetypal spartan woman and entirely blamed for all the dead of the Trojan war. Between line 1455-61 her name is played with repeatedly to talk of the destruction of ships, men and cities. (Homer by contrast is a lot more sympathetic in his depiction of Helen, making sure the audience knows that the gods, not Helen, are at fault e.g. Il. 3.163-4).

In Euripides' Andromache, Troades and Orestes Helen is sexual manipulative and controlling of her Spartan husband, Menelaus. Again, it is often repeated that she is not only a spartan but a typical Spartan woman. It's worth reading the speech by Peleus in Andromache. It starts with calling Helen a whore, then all spartan women whores, then describing their clothes as whoreish and then goes on into how Spartan men are weak.

This is the main theme through descriptions of Spartan women. It is not that the Spartan women are strong, it is that the Spartan men are weak (something an Athenian audience may have appreciated as they were loosing the war) . The same can be said of comedy, of the writings of Xenophon, Plato or Aristotle (with all sorts of caveats attached that I could go into if anybody wanted me to).

The best article on the topic (I haven't looked at the topic for 5 years so there is sure to be a lot more now) is Ellen Millender's (1999) Athenian Ideology and the Empowered Spartan Women.

There is also a good article by Paul Cartledge about modern feminist appropriations of Spartan women: (1981) Spartan Wives: Liberation or Licence?

Ancient sources regarding Spartan society can be shaky because most of them are not from Sparta itself.

However, what that tumblr post describes is definitely the classically accepted view of women in Spartan society. And while the ancient sources might not be completely accurate, it is pretty obvious that women did have more rights in Sparta and it is pretty easy to see why that was the case - namely, a society where men spend vast amounts of time fighting, training in state-run military barracks to fight, or dying whilst on campaign, will tend to require putting women in positions of greater importance.

However, quite a few of the people who like to glorify Sparta (pop historians, people who makes posts on tumblr like this) like to conveniently gloss over the fact that Spartan society was only sustainable due to its vast amount of people called helots, who made up the majority of the population of Sparta's domains. Spartan women who were citizens had those rights. If you were a helot woman, man, or child, you lived in a system that could generously be described as serfdom and really was at times quite closer to slavery. As I mentioned earlier, most of our sources are non-Spartan and a lot of them describe situations involving the treatment of helots that are a bit fanciful, but it is painfully clear that the helots were brutally oppressed and significantly mistreated, leading to numerous helot rebellions that significantly endangered the Spartan elite. Now, helots were not citizens so it is slightly understandable why people don't talk about them when they talk about "Spartans". However, talking about Spartan society, particularly about Spartan politics and hierarchy, without talking about the helots (and the other non-citizen classes) is a bit like talking about feudal Europe and not making mention of the serfs/peasants or talking about Spanish North America and not making mention of the slaves.