anything that causes someone to feel that they have more intrinsic worth than someone else.
For the rich, that often is their riches. ("I have more because I'm smarter, more educated, more frugal, more righteous . . ." In other words, "I'm better than the poor, because . . .")
For the poor, that often is their lack of riches. ("I have less because I care more about the important things, am less caught up in unrighteous pursuits of worldly accomplishment, am more willing to spend time with my family and sacrifice financially to do so, more righteous . . ." In other words, "I'm better than the rich, because . . .")
Pride has NOTHING to do with how much someone has financially, at the most basic level.
Sharing what you have with others can be a way to break down pride, in that it can be a source of humility and letting go of one's attachment to and self-definition from financial status. Seeing the actual benefits of sharing what you have is a much better antidote than "just giving" for many people, since it fills a basic human need to see the benefits of what we do. However, being able to give without seeing the actual benefits because of faith that the contribution actually is helping someone else is the ultimate form, in my opinion.
Tithing does that, in a very real way. We "know" the money is going to help build meetinghouses and temples in areas where we can't see the actual people who are benefited (less so now in our internet age, but still in concept) - and even the poorest can feel that sense of accomplishment (the type of pride that Pres. Uchtdorf mentioned is fine to feel) from their giving. It's important that they feel like a contributing part of the kingdom. (Thus, much of the crisis of faith over tithing for many people comes from lack of confidence in how tithing dollars are used - either out of a sense that "I know how MY MONEY should be spent," or simply not being able to see how it is spent, since "faith" in matters like this is hard to accept.)
For the poor, going to the Bishop for assistance helps break down the type of pride that says, "I'm my own person. I don't need help. My kids might miss meals and wear clothes with holes in the them, and my wife might go without basic necessities in some cases, but that's fine because I'm self-reliant and won't take help from anyone," - with an implied, "Damn it!!" at the end and a foundation of scorn for those who accept help from others. It often carries scorn for those who would help, and it generally carries scorn for those who accept help. That is pride at its most elemental level.
So, paying tithing helps provide "good pride" and can lead to getting rid of "bad pride" - including in those situations where people hoard their own money and don't share because, as King Benjamin said, they justify not sharing by assuming they would if they could. The point is that ALL "can" share and receive from sharing - but that sometimes has to be manufactured temporarily (or even long-term) for some people in some situations in order to accomplish BOTH objectives and not just one of them.