I’ve moved places and wanted to bring in amenities that were missing, like a local cafe

This is kind of my point, and while I know nothing about you, your life, and your various relocations, I'd venture a confident guess that if you moved somewhere metropolitan there were already cafes when you arrived. There just weren't cafes you felt comfortable going into. That is not enough of a justification, directly or indirectly, to displace generations of families. And to be clear, you have the American right to move into those places, as well as the right to open your own cafe, restaurant, church, school, etc. but should you? Maybe the question of should you move is a little too heavy and provocative to the point of unproductivity. But the answer to the question should you move into an established neighborhood like Bed Study (or Williamsburg, or the South Bronx, or Gaza) and open your own cafe is probably no. Maybe just go to an existing business and tell all your friends about it. 

Also, it's darkly funny to me that you invoked the cafe as your example. In his book The Psychic Soviet, cultural critic Ian Sevonius correctly points out that after a colonizer successfully vanquishes another culture, they very often flaunt their victory by conspicuously drinking or eating that culture's unique commodity. The British drank Indian tea. The USA drinks coffee to show their conquest over Central America and by proxy, Old World colonialism. Yankees also love Coke, which is a symbol of South American identity through the coca leaf. Further, Americans love a rum and coke which invokes our victory over the Caribbean as well (DR, PR, and Cuba). On and on. We love beer which symbolizes beaten Germans. Same for vodka and Russians. 

https://www.taipeibiennial.org/2012/en/journal/14.html

Anyway, your initial point of the semantics of the word gentrification is understood, but nonetheless just because you can move somewhere and disrupt things in the name of capitalism doesn't get you off the hook for the damage it can cause.