The second probe shows a faster response because it is peaked.
Test the oscilloscope itself without the probes by attaching the pulse generator directly to a vertical input. If you have to use a cable or adapters, then connect it through a 50 ohm feed-through adapter directly at the oscilloscope input.
If you are going to be messing around with probes and testing oscilloscopes, then a feed-through adapter is a good investment anyway.
https://www.amazon.com/Rigol-ADP0150BNC-Ohm-Impedance-Adapter/dp/B01C3EFUF0/
https://www.amazon.com/Zyyini-Adapter-Through-Terminator-Device/dp/B07SD3TC51/
It took a while for the feed-through adapter to arrive, but now it has and I tested the scope with the pulse generator directly attached to the scope inputs. Both channels appear to show a rise time of something like 1.4-1.5 ns, which, if my math is correct, comes to something like 470 MHz. I also tested again with the various probes, with similar results (in the range of 1.5-1.6 ns).
Attached is an image from one of the channels with the pulse generator directly attached through the terminator-adapter, with the timebase set at 0.01 us/div (10 ns/div) and using the 10x mag switch to give 1 ns/div. The other channel, and the images using the probes, look substantially similar (there's slightly more ringing with the probes in place, and the cheap P2200s actually seem to perform slightly better than the original Tek P6075As, but the rise times are basically the same to within 0.2 ns. At least I'll have no further hesitation about ordering cheap probes off Amazon).
I've been using the scope for a micro-processor project for a couple weeks, and I'm getting more confident in its capabilities, as well as its shortcomings. The channel 2 selector switch needs a cleaning pretty badly, as do both vertical position POTs and the beam intensity POT, and there's some kind of crosstalk between the channels when one of them is set to the highest sensitivity. Also, I'm pretty sure that the timebase has drifted a bit since it was last calibrated (in Dec. 1979!), which only adds to my uncertainty about the actual rise time, but otherwise this old thing is a real trooper and a pleasure to use.
I've ordered a "for parts" 475 off eBay, and it will arrive in a couple weeks. My intention is to keep that around for spare parts, but I'm going to open it up, try to diagnose it, and just get a feel for what it will be like refurbishing my father's old scope. I have hesitated to open up his scope without a pressing reason because I don't want to make any stupid mistakes with it, but a stranger's non-working scope is completely fair game.