The wavyness of your measurement using a probe is due to bad impedance matching. Generally, I wouldn't use alligator clips above 1 MHz if you want good signal fidelity. Use a BNC cable instead, preferably with a 50 Ohm termination (if the source is 50 ohms). Or, you could cut a BNC cable, and probe the cable within a few cm of the 10 MHz output (it's the length of the line between the source and the mismatch that allows reflections to create large ripples; reducing the length will make the standing waves smaller). The first measurements looked much better.
The SDG2042X's datasheet claims its output is 50 ohm, with a 3.3Vpp amplitude into Hi-Z. Your oscilloscope (DSO-X 2012A) doesn't have a 50-ohm load, so you should provide an external termination by using an in-line feedthru terminator (like Thor T4119). This would be attached directly at the oscilloscope's input. The amplitude should be halved with the termination, to around 1.3Vpp.
The signals themselves look OK, but not great. You may be able to get a rough idea of the period-to-period jitter using your scope. Use the horizontal trigger/delay knob (top of the scope, small knob), to zoom into the rising edge AFTER the one where the scope trigger (so a delay of 100ns). Turn on infinite persistence mode and get a feel for the width of the acquired signal, which would be the peak-to-peak period variation of your source. Since your oscilloscope samples at 2 GSa/s, I'd expect you to be able to measure it with a precision around a few hundred picoseconds. You may be able to zoom into an edge much after the trigger edge (like 1 second later) to get an idea for the longer-term jitter, too, though the oscilloscope's timebase jitter may start to dominate. You can also do some measurements by using edge delay measurements, but you may have to zoom out enough that your sampling rate decreases too much. If you acquire the data onto a computer and interpolate between sample-points, I think you could get a quite good idea of the jitter's magnitude and make pretty plots.
Post your results for the jitter if you get a chance. It's not the most advanced way, but it should give you an idea of its performance.
Often square waves are desirable for clocks because the signal transitions more quickly through the logic theshold value. Close to the midpoint, a sine wave changes more slowly than a square wave would, which lets more jitter into the system. Many good quality oscillators are clipped sine-waves or square waves, for this reason.
EDIT: And now I'm looking into Keysight's N8900A software which would provide offline jitter analysis, but OMFG, it's a >3 GiB download and costs US$8500, and all I wanted is to look at the user manual.