I believe ESRI would be the best people to talk to about this. I've setup Collector a few different times and it's a really cool product, especially when using feature services directly from ArcGIS Server, but there are a lot of different factors that could be causing you problems. Without having your map or data it's impossible to tell you what you're doing wrong.
One thing you may be missing is that when you first start the app and login, you need to enter the URL of your organization, for example:
organization.maps.arcgis.com
And then enter your username/password that is associated with the organization. This username also needs to be set to either an Administrator or Publisher role.
This ESRI post seems to be exactly what you're asking:
http://forums.arcgis.com/threads/89386-Please-use-an-organizational-account-error-on-collector
kylie from ESRI:
There are two different accounts that you can get for no cost. One is
a public account. Public accounts are not supported in Collector. The
other is a 30-day free trial of an ArcGIS subscription. Those are
supported with Collector. You will need to set up your organization,
and then you will be able to log in with the account tied to that
30-day trial. By set up your organization, it is not paying for it.
When you create the free trial you will get email that explains what
you need to do.
What's all in your map?
Are you adding a shapefile? Did you make sure to check "Publish this file as a feature layer"? The editable layer has to be a feature service. Also, you need to turn editing on for the feature. I'll add some screenshots.
If editing says Disabled, you need to click Edit and at the bottom of the page turn it on.
Go into the map on AGO and make sure you can edit it there.
You only need to enabled Feature Access on the service when publishing and have some variant of Add/Update/Delete enabled.
Also you need to share the map and layers to a group that your other org members are a part off. The app will only display maps shared across groups that you are a member of or maps that exist under your My Content. These maps also need to have an editable layer included to be picked up by the app when it accesses your available content.
Best Answer
I have created a few web maps for Collector for ArcGIS and I always do it the same way.
I always start off by creating a file geodatabase through ArcMap. (Catalog and Pro will work the same) I make sure to finish all of my symbology, domains, and subtypes before publishing it as a web layer.
Go to File-> Sign In...-> Sign in with your ESRI account.
Go to File-> Share AS-> Service...-> Publish a Service-> Choose a connection: My Hosted Services('Your organization') Service Name: 'Title'. I change my capabilities to Feature Access, not Tiled Mapping. Set up your feature access operations to preferences. I normally share it with Everyone
Then you will want to publish this service.
Create it as a web map online and it works without having to even touch the settings.
A suggestion that I would consider is trying to share it with a group that you are a member of.