OK ... so hopefully ("fingers crossed") this is my final post on this dreaded topic. I will start by saying ... I got past the "open_basedir restriction in effect" error successfully (still have not finished the install but am hoping the rest will go smoothly. I have spent 3 days (maybe 4 or 5 days ... it is all a blur at this point) with what seems to be a simple problem. My final solution was simple, but it does not make sense way this last step was required according to the Plesk Admin user guide link I posted previously.
First, if you are new to Plesk, make sure you understand the difference between Service Plan, Subscription and Customer (I am assuming you are logging into the Admin side of Plesk, and that you are NOT logging in as a customer in the customer control panel). If you are using a 1and1 virtual host setup, by default there is a Custom Service Plan tied to the "no name" domain name 1and1 gives you by default (e.g., us1234.dynamiccloudserver.info (Custom)). This Custom Service Plan can not be adjusted.
You need to create a new service plan and, if you are the only one hosting websites on your server (i.e., you are not reselling hosting packages), just give this Service Plan unlimited everything (well, at least that is what I did ... figure it did not matter).
If you did not already do so, add yourdomain.com to a subscription by going to Hosting Services>Subscriptions>Add New Subscriptions ... the page should say Adding Own Subscription. On this page, you can select the Service Plan you already created (not the Custom plan that can not be changed).
If you already had added yourdomain.com to a subscription, go to Hosting Services>Domains, click on yourdomain.com, then click on the Change Plan button. Select the plan on the New Service Plan dropdown menu and click OK.
Now that you hopefully have the correct Service Plan attached to yourdomain.com Subscription, you can do what was already stated above with the custom PHP settings in the Plesk control panel.
So here was my final problem and how I fixed it. First, being logged in under Plesk Admin control panel, go to Hosting Services>Service Plan>Hosting Plan (tab), under the Plan Name header, click on the name of the Service Plan you created earlier. Now you should be at Home > Service Plans > Editing MyServicePlanName. Click on the PHP Settings tab. Under the Perfomance Settings, go down to "open_basedir", click on Enter Custom Value buble and enter /var/www/vhosts/yourdomain.com/httpdocs/:/var/www/vhosts/yourdomain.com/moodledata:/tmp:/proc:/usr/bin . Click on the Update & Sync button at the bottom of the page. At this point, I restarted my web service under Service Management>Tools & Settings>Services Management by clicking on the orange icon to the far right on the Web Server (Apache) line. This still did not fix the problem for me.
I then went to Hosting Services>Subscriptions and clicked on the Open in Control Panel on the far right of the mydomain.com line. Once into the mydomain.com control panel (you are now logged in as yourdomain.com subscription the Service Plan, instead of the overall hosting admin), click on the Websites and Domains tab. Now go down toward the bottom of the page under the Domain header, and click on mydomain.com.
You should now be on a page titled Hosting Settings for YourDomain.com. If you setup the Service Plan and Subscription correctly, you should have (at least) two tabs: General and PHP Settings. Click on the PHP Settings tab. As before, scroll down to "open_basedir" and click on Enter Custom Value bubble and enter the same /var/www/vhosts/yourdomain.com/httpdocs/:/var/www/vhosts/yourdomain.com/moodledata:/tmp:/proc:/usr/bin and click OK button at the bottom of the page. I also had a subdomain I was using to point at my /moodle directory, so had added it as a subdomain in MyDomain.com Subscription, and followed the same steps to Enter Custom Value for open_basedir in PHP Settings (just for good measure).
Finally, I got the install to work. Not knowing Plesk that well, maybe I could have just done the settings in the MyDomain.com Plesk control panel, but I did it in all of them. After reading the Plesk Admin User Guide I linked in the previous post, it seems to me that changing this setting only in the Plesk Admin login should have resulted in it working correctly, but it was not until I did it in the specific MyDomain.com PHP Settings, did it work.
Good luck and hopefully you don't hear from me on this issue again!! 