Do I need to prove that my spouse is using their right of free movement in order to receive a family member Schengen visa?

Unfortunately your spouse, being resident in Pakistan, is not exercising their right of free movement and so you will not be able to travel with them (without applying for a regular visa). Your spouse would need to be living in an EU/EEA country other than the UK in order to be exercising their rights and thus to be able to travel freely with their family members.

The crucial text is at the top of the picture you posted: "by living/working in another EU/EEA country ... than he/she is a national of."

Some explanatory text courtesy of the European Commission can be found on page 6 and is copied below - see the very last line:

At the suggestion of @phoog I checked other languages:

The other answer has been voted down because there was a long discussion of whether a UK citizen visiting the Netherlands for a holiday is "exercising their EU rights of free movement" or not, and there is absolutely no agreement. So this other answer may be totally correct or totally wrong, but people can't agree.

Fact is that you have the legal right to move to the Netherlands with your partner permanently (well, until October 31st if nothing changes), rent or buy a house, get a job and so on. The biggest reason why countries tend to refuse visitor visas is that they fear the person who enters legally for a holiday will then stay longer illegally. They shouldn't fear that in your case, because if you intended to stay longer, you could do so perfectly legally and there would be no need for you to do it illegally.

So you might consider applying for a visitor visa, and add a note pointing this out, as evidence that you will be returning home.

Since there is no posted answer espousing the view that "a UK citizen visiting the Netherlands for a holiday is 'exercising their EU rights of free movement,'" I will post one.

Obviously that is true only while the UK continues to participate in the EU's freedom of movement scheme, which currently seems like it will be until the end of this year. Nonetheless, the question will continue to be relevant for people with the nationality of other EU and Schengen countries.

This answer will discuss the language that gave rise to the question in the first place as well as the language on VFS Global's page for EU/EEA/Swiss family member applications in Pakistan.

First, the question cites a checklist that says in part

6. Proof that this family member is using or has recently used his/her right of free movement.
   For example: a proof of registration or a residence permit of your family member (as an EU/EEA or Swiss citizen) in the Netherlands.
   If you are traveling to another Schengen Member State represented by the Netherlands for visa issuance, proof of registration / residence permit of your family member in the Schengen country of destination.

Note that the registration/residence permit is offered as an example. This does not exclude the possibility that acceptable proof could be a flight reservation or other evidence showing that the EU/EEA/Swiss family member either will arrive in the Netherlands with the applicant or will be in the Netherlands when the applicant arrives.

Second, the page linked above lists eligibility criteria. These criteria are somewhat oversimplified, but they nonetheless show that you should be eligible:

Eligibility

You are eligible if you can prove objectively that all of the following requirements are met:

1. You are a family member of an EU/EEA/Swiss national.
   (this includes a spouse, registered partner, child who is under 21 or a depending family member); and
2. you are accompanying the EU/EEA/Swiss national or planning to join him/her; and
3. this EU/EEA/Swiss national is travelling to or is residing in another member state than that of which he/she is a national.

The first two points are not disputed. Note that the third point says "traveling to or residing in" (emphasis added), so the spouse of a non-Dutch EU citizen who is traveling to the Netherlands qualifies for this type of visa regardless of anyone's place of residence, and regardless of the reason for the trip. We can therefore conclude that "[an EU] citizen visiting the Netherlands for a holiday is 'exercising their EU rights of free movement.'"

(The same is true of any qualifying family member of any national of Switzerland or any EU or EEA country other than the Netherlands. Furthermore, this is not mentioned because of the oversimplification, but there are circumstances under which it can also apply to citizens of the Netherlands and their family members, namely if the Dutch citizen resides in another EU or EEA country or Switzerland.)

Anyone who encounters resistance on this point may want to raise the issue with SOLVIT (in English or Dutch, among other options).