There is no "secure" way to do this unfortunately. Field-level access is not something you can restrict.
Having said that, you can create alternate forms and simply show/hide the links for these forms using audience targetting.
Next, you want to setup manual links using webparts that have audience targetting set to the respective users/groups that get access to each form.
Link 1, targets users a, b, c, points to NewForm1.aspx
Link 2, targets users d, e, g, points to NewForm2.aspx
To add these links, use content editor webparts or script editor webparts. In the webpart itself just add an anchor tag like this
<a href="urlToNewForm1.aspx">Form 1</a>. Style it using CSS if you're familiar.
The Newform1.aspx can be added by opening the site/list in SharePoint Designer. Or you could probably use the windows explorer view in IE to copy-paste a newform.aspx into the Forms folder of the list. I recommend SPD though.
Another secure option would be to code a provider hosted solution that uses elevated permission and restrict edit access to the list through the provider hosted app only.
Hope this helps!