Generalize prediction cache. (#8783)

* Extract most of the functionality into `DMatrixCache`.
* Move API entry to independent file to reduce dependency on `predictor.h` file.
* Add test.

---------

Co-authored-by: Philip Hyunsu Cho <chohyu01@cs.washington.edu>

src/c_api/c_api.cc

@@ -12,11 +12,11 @@
 #include <vector>
 
 #include "../collective/communicator-inl.h"
+#include "../common/api_entry.h"  // XGBAPIThreadLocalEntry
 #include "../common/charconv.h"
 #include "../common/io.h"
 #include "../data/adapter.h"
 #include "../data/simple_dmatrix.h"
-#include "c_api_error.h"
 #include "c_api_utils.h"
 #include "xgboost/base.h"
 #include "xgboost/data.h"