As far the terms:
Bass Tone Glossary | TalkBass.com
looking at this video:
Im hearing “punch” but it’s not necessarily “bright,” apart from the fresh roundwound sound. It sounds like a Jazz (which it is,) maybe a little more of the bridge pickup. Maybe a little dirt and reverb. (I’m sure I’ll get corrected on my assessment

)
I’m not huge on replicating bass tones, but whatever you do, learn about EQ. Whatever amp you get, learn what the knobs control, and spend time playing with them. What does it sound like cranking up the low mids? Ok, and cutting them? Really get a taste of what the knobs do, and you will naturally gravitate towards the “bass tone you like.”
of those heads: Tim and Jason are both knows for using Ampeg. But a PF sounding like an SVT is debatable, and I’ve read enough bad reliability reports on those to scare me off (many TBers vouch for them though.) The Peavey is going to be cleaner sounding, but I’ve never been let down by a piece of Peavey gear.
good luck!