Is this a safe way to measure mains power on an oscilloscope?

That looks like a good safe project to me. You might consider checking the voltage between the 9.9 V terminals and earth. You will probably find some random voltage perhaps even 115 volts. Connect a 115 volt light incandescent bulb between the 9.9 V terminal and earth. With that connection, the voltage across the bulb should drop to nearly zero. That should demonstrate that there is some small leakage current through the transformer due to capacitance between the primary and secondary winding.

From your description, it sounds like you've made a safe isolated source of 9 VAC.

To verify that the output is isolated from the AC Mains, you could measure (with the thing unplugged!) the resistance between the mains terminals and your output terminals - there should be no continuity (inifinite resistance) between the mains pins and the 9 VAC jacks.

and found a lot of information, some of it contradictory.

Some people take a more paranoic approach to safety than others. Also some scopes are more suitable for mains measurements than others.

In general there are three main issues when using a scope directly on the mains.

1. Scopes are normally referenced to mains ground, so if you hook your probe ground lead to mains neutral you will create a neutral to earth fault which will trip RCDs/GFCIs and could potentially melt wires. You can get around this by just measuring with respect to ground. It's probably close enough and you can always measure the neutral and subtract.
2. Voltage ratings, many scopes have too low a voltage rating for mains power. You can work around this with an attenuated probe.
3. The thing that many people forget, overvoltage categories. Many scopes are only "Cat 1" which are not suitable for direct connection to the mains. For connection to mains power from a domestic outlet you want at least "Cat II". To fix this you really want an isolated probing soloution.

Your transformer solves all of these issues, and it is safe to connect your scope to it's output (secondary circuits of normal mains transformers are considered CAT 1). However it may well distort your results.

The proper solution is a high-voltage differential probe, but such a device is out of the price range of the casual tinkerer.