Let's try to share the same logic with desktop if possible.

Frome pref_metrics_service.cc

  UMA_HISTOGRAM_BOOLEAN("Settings.ShowHomeButton", show_home_button);
  if (show_home_button) {
    UMA_HISTOGRAM_BOOLEAN("Settings.GivenShowHomeButton_HomePageIsNewTabPage",
                          home_page_is_ntp);
  }

  // For non-NTP homepages, see if the URL comes from the same TLD+1 as a known
  // search engine.  Note that this is only an approximation of search engine
  // use, due to both false negatives (pages that come from unknown TLD+1 X but
  // consist of a search box that sends to known TLD+1 Y) and false positives
  // (pages that share a TLD+1 with a known engine but aren't actually search
  // pages, e.g. plus.google.com).  Additionally, record the TLD+1 of non-NTP
  // homepages through the privacy-preserving Rappor service.
  if (!home_page_is_ntp) {
    GURL homepage_url(prefs_->GetString(prefs::kHomePage));
    if (homepage_url.is_valid()) {
      UMA_HISTOGRAM_ENUMERATION(
          "Settings.HomePageEngineType",
          TemplateURLPrepopulateData::GetEngineType(homepage_url),
          SEARCH_ENGINE_MAX);
      rappor::SampleDomainAndRegistryFromGURL(
          g_browser_process->rappor_service(), "Settings.HomePage2",
          homepage_url);
    }
  }

Granted, we don't store it via chrome prefs (since we never want it to sync), but maybe I can pull that out and call it with our values.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/42da7e24e22f88b02bd4dc64af9206b6ab40d84c

commit 42da7e24e22f88b02bd4dc64af9206b6ab40d84c
Author: tedchoc <tedchoc@chromium.org>
Date: Wed Jun 01 03:56:49 2016

Log some information about the state of the homepage on Android.

BUG= 612653 

Review-Url: https://codereview.chromium.org/2006023002
Cr-Commit-Position: refs/heads/master@{#397046}

[modify] https://crrev.com/42da7e24e22f88b02bd4dc64af9206b6ab40d84c/chrome/android/java/src/org/chromium/chrome/browser/DeferredStartupHandler.java
[modify] https://crrev.com/42da7e24e22f88b02bd4dc64af9206b6ab40d84c/chrome/android/java/src/org/chromium/chrome/browser/metrics/LaunchMetrics.java
[modify] https://crrev.com/42da7e24e22f88b02bd4dc64af9206b6ab40d84c/chrome/browser/android/metrics/launch_metrics.cc
[modify] https://crrev.com/42da7e24e22f88b02bd4dc64af9206b6ab40d84c/chrome/browser/prefs/pref_metrics_service.cc
[modify] https://crrev.com/42da7e24e22f88b02bd4dc64af9206b6ab40d84c/chrome/browser/prefs/pref_metrics_service.h

Your change meets the bar and is auto-approved for M52 (branch: 2743)

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This issue has been approved for a merge. Please merge the fix to any appropriate branches as soon as possible!

If all merges have been completed, please remove any remaining Merge-Approved labels from this issue.

Thanks for your time! To disable nags, add the Disable-Nags label.

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The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e

commit f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e
Author: Ted Choc <tedchoc@google.com>
Date: Fri Jun 10 17:37:42 2016

Log some information about the state of the homepage on Android.

BUG= 612653 

Review URL: https://codereview.chromium.org/2054983004 .

Review-Url: https://codereview.chromium.org/2006023002
Cr-Original-Commit-Position: refs/heads/master@{#397046}
Cr-Commit-Position: refs/branch-heads/2743@{#318}
Cr-Branched-From: 2b3ae3b8090361f8af5a611712fc1a5ab2de53cb-refs/heads/master@{#394939}

[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/android/java/src/org/chromium/chrome/browser/DeferredStartupHandler.java
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/android/java/src/org/chromium/chrome/browser/metrics/LaunchMetrics.java
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/browser/android/metrics/launch_metrics.cc
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/browser/prefs/pref_metrics_service.cc
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/browser/prefs/pref_metrics_service.h

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e

commit f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e
Author: Ted Choc <tedchoc@google.com>
Date: Fri Jun 10 17:37:42 2016

Log some information about the state of the homepage on Android.

BUG= 612653 

Review URL: https://codereview.chromium.org/2054983004 .

Review-Url: https://codereview.chromium.org/2006023002
Cr-Original-Commit-Position: refs/heads/master@{#397046}
Cr-Commit-Position: refs/branch-heads/2743@{#318}
Cr-Branched-From: 2b3ae3b8090361f8af5a611712fc1a5ab2de53cb-refs/heads/master@{#394939}

[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/android/java/src/org/chromium/chrome/browser/DeferredStartupHandler.java
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/android/java/src/org/chromium/chrome/browser/metrics/LaunchMetrics.java
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/browser/android/metrics/launch_metrics.cc
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/browser/prefs/pref_metrics_service.cc
[modify] https://crrev.com/f3412455b704ba36ac9e3bdbbe1d2b5d415b8d8e/chrome/browser/prefs/pref_metrics_service.h

Issue 540831 has been merged into this issue.