analyze format=json select * from one_k t1, t2 where t1.a+t2.a<1000;

ANALYZE: {
  "query_block": {
    "select_id": 1,
    "r_loops": 1,
    "r_total_time_ms": 5818.7,
    "table": {
      "table_name": "t1",
      "access_type": "ALL",
      "r_loops": 1,
      "rows": 1000,
      "r_rows": 1000,
      "r_total_time_ms": 21.695,
      "filtered": 100,
      "r_filtered": 100
    },
    "block-nl-join": {
      "table": {
        "table_name": "t2",
        "access_type": "ALL",
        "r_loops": 1,
        "rows": 1000,
        "r_rows": 1000,
        "r_total_time_ms": 18.274,
        "filtered": 100,
        "r_filtered": 100
      },
      "buffer_type": "flat",
      "buffer_size": "128Kb",
      "join_type": "BNL",
      "attached_condition": "((t1.a + t2.a) < 1000)",
      "r_filtered": 50.05
    }
  }
}

Here we see that t2 has r_loops=1, which means t2 was accessed one time.
Is this sufficient to indicate that the buffer was filled only once?

My suggestion for the first approach: let ANALYZE show another field:
r_buffer_used. It should print

  r_buffer_used: size

when the buffer was used one time, and

  r_buffer_used: "multiple times"

when the buffer was used multiple times.

I have a similar solution for ANALYZE+filesort().  If filesort was invoked once,
it prints:

  r_sort_passes: nnnn,
...

and if this is a subquery and filesort was run many times, it prints

  r_sort_passes: "(varied across executions)"