Hi @Lauren_Johnson and welcome to the community!
If I understand correctly, you have multiple projects, each with a certain project type, that can link to 1 or more activities. Those activities only have 1 engagement type. You want to know per activity, how many project types are linked to it per project type?
If I’m correct, you can use a conditional rollup field, where you choose “Only include linked records from the [projects table] that meet certain conditions”. In that condition, you can choose the project type you want to rollup. You can create a rollup field per project type.
Or, if you don’t need the numbers for anything else, you could maybe also use the pivot table extension.