Do I need a transit visa in Amsterdam going from Philippines to Greece, if travelling on a (government) official passport?

An airport transit visa (ATV) does not make sense in this scenario, that much seems clear.

Article 3 of the Schengen Visa Code (emphasis mine) provides that

1. Nationals of the third countries listed in Annex IV shall be required to hold an airport transit visa when passing through the international transit areas of airports situated on the territory of the Member States.

But if you are flying from Amsterdam to Greece, you will have to leave the international transit area. And the Philippines is not on the Annex IV list. On a purely legal level, this article therefore clearly does not apply and there is no basis to require or issue an ATV here.

In practical terms, all passengers who want to take a flight to other places in the Schengen area have to go through an “external border check” in Amsterdam and you can't do that with an ATV. There are no facilities for sterile transit in this case, the assumption being that people who can enter their final destination in the Schengen area are generally allowed to enter the country where they are transiting as well. In this situation, Philippines citizens with regular passports would therefore need a type C “uniform” Schengen visa like they would to enter either Greece or the Netherlands.

Incidentally, the same article also mentions this:

5. The following categories of persons shall be exempt from the requirement to hold an airport transit visa provided for in paragraphs 1 and 2:

[…]

(e) holders of diplomatic passports;

Even if your passport is not a diplomatic one and that last provision is not relevant, there are several different reasons why an airport transit visa is not applicable. So if you need something, it's going to be something else than an ATV (probably a Schengen visa, with or without territorial limitation). You should check what the requirements are to enter the Netherlands on your passport.

First of all, for border purposes the Netherlands (Amsterdam) and Greece are in the same country: the Schengen area, with a common visa.

As such, while the Manila-Amsterdam leg will be international, the Amsterdam-Athens leg is domestic, and immigration will be cleared in Amsterdam.

However, the visa requirements for entering Schengen through different member states do differ for diplomatic and official passport holders.

The following info is from Timatic, the database used by airlines.

For the Netherlands:

Visa required.

For Greece:

Visa required, except for Nationals of Philippines with an official passport for a maximum stay of 90 days.

In other words, had you entered Schengen at an airport in Greece (or Austria, Denmark, Estonia, Finland, Germany, Hungary, Italy, Lithuania, Malta, Poland, Slovakia, Slovenia, Spain or Sweden) you would not have needed a Schengen visa.

However, if entering at an airport in the Netherlands, unfortunately you do need a (Schengen type C) visa. An airport transit visa will not do, as you're not staying in the international section of the airport in Amsterdam.