Tested the issue on reported chrome version 63.0.3239.84 and on the latest canary 65.0.3309.0 using windows 10 with the below mentioned steps.
1. Launched chrome and opened a web page
2. Opened Dev tools and clicked on application
3. Then to Storage>web SQL

We are unable to find any names with mydatabasename or similar to that in order to perform a query. Attaching the screen cast of the same.

@Reporter: Could you please check the screen cast and let us know whether we need to have a database connection in order to check this. It would be highly helpful if provided with sample SQL queries to test from ET end.

Thanks!  

Deleted: 798785.mp4 1.8 MB

	798785.mp4 1.8 MB View Download

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/4b4a5822e6c6c522db2d3803c4632f9346f92a1e

commit 4b4a5822e6c6c522db2d3803c4632f9346f92a1e
Author: Eugene Ostroukhov <eostroukhov@chromium.org>
Date: Sat May 12 01:30:42 2018

DevTools: fix WebSQL query view issues

1. Focus the prompt when the view is focused.
2. Do not hide the query while waiting for the results.
3. Show query in history if it does not produce results
   (e.g. DELETE ... FROM)

Bug:  798785 
Change-Id: I7cc86301256e0370f03282e044bb59fef0a26dbe
Reviewed-on: https://chromium-review.googlesource.com/1056072
Reviewed-by: Andrey Kosyakov <caseq@chromium.org>
Commit-Queue: Eugene Ostroukhov <eostroukhov@chromium.org>
Cr-Commit-Position: refs/heads/master@{#558085}
[modify] https://crrev.com/4b4a5822e6c6c522db2d3803c4632f9346f92a1e/third_party/blink/renderer/devtools/front_end/resources/DatabaseQueryView.js

@Eugene Ostroukhov: Could you please let us know if we need to have a database connection in order to reproduce/Check the issue(i.e., as we are unable to reproduce it from our end-comment#1). And please help us in verifying the fix.

Thanks!

Sorry guys/gals. I thought my original post was self-explanatory. I'm not
sure if I'm using the correct verbiage to describe the problem. It should
be simple enough.

I see a ">" there in the view awaiting my query on the first open of the
view. I click near it and the see the vertical bar "|" appear. I type out
my query and hit enter. The query runs. If results are returned, I see them
and a new ">" entry point below. It's at this point that I can't seem to
get a new "|" by clicking near the ">", nor does typing anything seem to
work. It used to. I don't really understand why.

As well, I am frustrated by the lack of messaging after any query is run
that returns no results. This tool is not nearly as intuitive as most any
other database tool. I realize it's a simpler interface built into a
browser instance, but surely it can be improved.

Not sure what else to say or include to help you help me. :(