OP here. It's no EXPLAIN ANALYZE, but here are the EXPLAINs at least. The origin...

OP here. It's no EXPLAIN ANALYZE, but here are the EXPLAINs at least. The originals were averages of 10 runs. These EXPLAINs are just quick runs on my laptop, so no promises. :)

First query:

                                             QUERY PLAN
  -----------------------------------------------------------------------------------------------------
 Sort  (cost=3475858.60..3475859.56 rows=385 width=22)
   Sort Key: (count(DISTINCT time_on_site_logs.user_id))
   ->  GroupAggregate  (cost=3370756.94..3475842.06 rows=385 width=22)
         ->  Sort  (cost=3370756.94..3405784.03 rows=14010837 width=22)
               Sort Key: dashboards.name
               ->  Hash Join  (cost=44.90..558337.80 rows=14010837 width=22)
                     Hash Cond: (time_on_site_logs.dashboard_id = dashboards.id)
                     ->  Seq Scan on time_on_site_logs  (cost=0.00..365621.11 rows=14016911 width=8)
                     ->  Hash  (cost=29.40..29.40 rows=1240 width=22)
                           ->  Seq Scan on dashboards  (cost=0.00..29.40 rows=1240 width=22)

(10 rows)

Second Query:

                                             QUERY PLAN
  -----------------------------------------------------------------------------------------------------
 Sort  (cost=2709630.36..2709631.09 rows=291 width=26)
   Sort Key: (count(DISTINCT time_on_site_logs.user_id))
   ->  Merge Join  (cost=2604392.06..2709618.45 rows=291 width=26)
         Merge Cond: (time_on_site_logs.dashboard_id = dashboards.id)
         ->  GroupAggregate  (cost=2604392.06..2709521.80 rows=291 width=8)
               ->  Sort  (cost=2604392.06..2639434.34 rows=14016911 width=8)
                     Sort Key: time_on_site_logs.dashboard_id
                     ->  Seq Scan on time_on_site_logs  (cost=0.00..365621.11 rows=14016911 width=8)
         ->  Index Scan using dashboards_pkey on dashboards  (cost=0.00..87.00 rows=1240 width=22)

Third Query:

                                                   QUERY PLAN
  -----------------------------------------------------------------------------------------------------------------
 Sort  (cost=436381.86..436382.36 rows=200 width=26)
   Sort Key: log_counts.ct
   ->  Hash Join  (cost=436308.72..436374.22 rows=200 width=26)
         Hash Cond: (dashboards.id = log_counts.dashboard_id)
         ->  Seq Scan on dashboards  (cost=0.00..29.40 rows=1240 width=22)
         ->  Hash  (cost=436306.22..436306.22 rows=200 width=12)
               ->  Subquery Scan on log_counts  (cost=436302.22..436306.22 rows=200 width=12)
                     ->  HashAggregate  (cost=436302.22..436304.22 rows=200 width=4)
                           ->  HashAggregate  (cost=435705.66..435944.28 rows=23862 width=8)
                                 ->  Seq Scan on time_on_site_logs  (cost=0.00..365621.11 rows=14016911 width=8)