Hmm, I'd like more details if you don't mind sharing, because we might have a red herring here. All Infopath forms are Infopath forms. The difference is whether or not it's browser-enabled. Even when you select this checkbox in the Form Options, it can still be opened and used as a rich client form, but it will have some features removed/restricted. Even when this option is selected, it doesn't change the behavior in SharePoint unless you change the Advanced Setting in the forms library to default to "Open in Browser." Only then will the form default to the browser version, and that only works if you've browser-enabled the form itself. Regardless, this form can still be used in rich client mode by selecting "Edit in Infopath" from the pulldown menu in the library once created.
So, what I'm getting at is that it doesn't matter if this is a browser-enabled form or not - it should work either way. You didn't answer my initial questions, so I could only assume that you had not previously published the form as a template to a forms library. I'm not sure what the situation was, but I can tell you how to accomplish your filename goal:
1) Turn off the Save/Save As (Views as well) buttons in the Form Options > Browser tab.
2) Create a data element in your form named "Filename" or whatever you prefer. Make it a regular text field with no attributes, and don't place it on the canvas.
3) Create a Submit-type data connection back to your forms library. Allow overwrite if filename exists and specify the filename as the Filename data field created in step 2.
4) Place a plain button on your canvas and change the properties accordingly:
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Change the label to Submit (or whatever you prefer)
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Add these Rules in this order with these conditions:
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If Filename is not blank, then Submit to Data Connection from step 3 and Close
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If Filename is blank, then Set Filename to Formula (create your preferred unique filename here - something like concat(userName(),"-",now())), Submit to Data Connection from step 3 and Close
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What this does is create a unique filename automatically when submitted and closes the file cleanly with no prompts. You could also choose not to close and leave the form open if you prefer...or have two buttons side-by-side - one that closes and one that doesn't, which is in essence a 'save' button instead of submit. This also makes sure that any subsequent openings of this exact file get saved as the same filename each time, which means it overwrites the previous version each time it's changed. You want this, because otherwise you get a new filename each time the file is modified...assuming files get modified after initial creation.