This depends very much on how you measure things like what it means for a person to be part of the group you're envisioning when you say "the party where people [whatever]."
To better highlight what I mean, I'll relate political parties to sports teams. An American football team has 40-50 people wearing the jerseys and doing something on the field at some point during the season. Let's call our hypothetical team the Donkeys. Clearly, every single one of those 40-50 people (the candidates) are members of the Donkeys.
Then you have the coaching staff and other support staff (political consultants/campaign staff/canvassers). They're not part of the "team" in the sense that they don't play football wearing a Donkeys jersey, but they're clearly part of the team in the sense that they make up the rest of the operational unit whose objective is to train, equip, and enable the jersey-wearers to play the game on Monday night.
Then there's ride-or-die fans (registered Democratic voters), the folks who own season tickets, show up to VIP fan events, hold tailgate parties, and preach the gospel of how the Donkeys are the best football team to ever take the field. They're the folks trash talking the Elephants and their fans on Sportsbook. Are they part of the team? The team's ticket and advertising sales don't happen without them... and here's where politics and sports diverge. Most people who talk about sports don't mean the fans when they say "The team." But for political parties, registered voters are a huge part of the party's ability to operate - I feel they should be included, but you may not.
Finally you have the casual football fans (unregistered, registered independents, and nonvoters) who participate in the events but to a much lesser degree. They definitely don't count as part of the team in sports - but in the comments of the OP you talk about Democrats having richer 'voters.' An independent who votes for Biden is counted by some to be a Democratic voter (at least in that election). Should they count? I feel they should not, because they don't consider themselves to be Democrats.
With those priors declared, we can answer the question:
Based on data from 2016, 2018, and 2020, the answer is "probably not." I'll get to that 'probably' at the end.
Consistently, populations of those whose household income is below $30k/year or so strongly prefer to identify as Democrats over Republican - and a much larger portion of the Democratic party's registered voters fall into that category than is true for the Republican party.
At higher income levels, the Republican party enjoys a modest population preference (2-5 points). And a larger share of the Republican party's registered voters are in the $100k+/year income bracket than is true of the Democratic party.
By those numbers, the Democratic party is certainly not the "party of the rich." It's quite the opposite, it's strongest voter identification advantage is among the poorest - and the largest number of the poorest choose to identify as democrats than any other identity.
So what's up with party performance in wealthier districts? Well, if we include the 'casual fans' as part of the equation we get something like this.:
Which shows larger proportions of poor folks turning out for Trump than did for Clinton in '16, or Biden in '20.
This is where your sample of 2018 & 2020 is hurting you. Trump's politics and political performance are outliers against the historical trends - and as I've demonstrated, even against other indicators. Trump's entire campaign was aimed at mobilizing large numbers of disgruntled, poor, mostly white folks who were dissatisfied with the status quo. He promised chaos and disruption more than anything else - and to people who felt like they were losing under the status quo, that turned out to have strong appeal. But it's not the typical tack taken by Republican candidates who tend to favor Reagan's "if we help the rich businessmen, they'll share the wealth down to the workers" economic model. (Hence the laser focus on tax cuts.)
In the 2018 data I linked above, there's a note about how the 'dollar effect' where more income drives more Republicanism there's a tipping point where the ultra-wealthy will again begin preferring Democrats. Especially in the era of Trump, these voters are also being driven by Trump's promise of chaos and disruption - driven in the opposite direction is all. There's also the Hollywood cluster, and a slew of ultrawealthy philanthropists whose approach to wealth is nontypical. In general the ultrawealthy are outliers themselves and so not reliable as representative samples of anything.
Finally, it's important to note that income is only part of the picture. This data from Pew about the households making $100k+ shows just how much geography can play a role in all this, too.