The closest is now the "Research Universities" which are sorted into "high" and "very high" reserach activity. You'd have to compare to the old R1/R2 lists to see how it shakes out, but I think the Very high is roughly R1 and high is roughly R2.

In terms of moving up from say a high to very high, I think its definitely possible but really depends on what you're doing and where you're doing it. Some of the less research heavy institutions might not give you the time or resources to develop the high-powered research program you'd need for a truly "R1" place. But the variability within each group is such that individual psychology departments and programs within an "R2" might actually be stronger than an "R1" so don't get caught up in it too much.

Find a place that makes you happy personally and professionally and go with it.