Get your taxes done using TurboTax

does an e-File of 1040-X count as one of your five e-Files? 

I don't think so.

in any case, when maling,

Do not include your old 1040 nor your revised 1040 because the Form 1040-X reflects any changes there and becomes your new tax return.

Include any new/corrected tax document only if it shows federal tax withheld, and include in order only those Sequence Numbered tax forms and Schedules that changed or were added due to your amendment.
Attachment Sequence Number is on all the forms and Schedules that come after 1040.

If you are using TurboTax you have to do this by careful visual comparison. The program won't break out those pages for you.
If your amendment dropped a form,
you should include with Form 1040-X a new copy of the form(s) with -0- in the appropriate lines or explain why in Part II Explanation .
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If any forms specify an attachment, put all those attachments at the end in the same order as the forms they go with.
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Mail to the address shown for your location in the most recent Instructions for Form 1040-X.