What's the name of this visual search task?
Aren't you just looking for a conjunction search task?

Normally it works like this: objects are described in terms of their features (e.g., colour, shape, etc.). If you tell people to search for a target with a given colour AND shape, then this is a conjunction of features, and is termed a conjunction task.

Note that there are complex dynamics in terms of how this works, and the second link below discusses this at length.

Here's some links on this type of thing:

http://www.mendeley.com/research/guided-search-an-alternative-to-the-feature-integration-model-for-visual-search/

http://www.mendeley.com/research/visual-search-stimulus-similarity-2/

Here are examples of Feature Search vs Conjunction Search from Wikipedia:
Feature Search:



Conjunction search: