Copyright always applies - unless the very few exceptions the law defines.
Licenses are uni-lateral "contracts", thus means by authors / creators / programmers / ... to give other people permissions to use or distribute a programme (or other work), under certain conditions which are not automatically granted by the law.
The law defines a few exceptions to the protection of intellectual property, and the details vary a lot between jurisdictions. By the Berne convention, copyright protection ceases after a certain period has passed after the death of the creator. And there usually are certain limits imposed which allow citations or deny copyright protection for trivial works.
Given this, copyright law, of course, applies to all open source licenses - a license cannot superseed the law; it is a statement by the authors to grant users or recipients additional rights. An open-source license is - roughly-speaking - the software equivalent of a sign on your acre of land which reads "free camping, if you don't litter and don't bother others". Open-source means, that the freedoms granted to users include usually
- The freedom to run the program as you wish, for any purpose (freedom 0).
- The freedom to study how the program works, and change it so it does your computing as you wish (freedom 1). Access to the source code is a precondition for this.
- The freedom to redistribute copies so you can help others (freedom 2).
- The freedom to distribute copies of your modified versions to others (freedom 3). By doing this you can give the whole community a chance to benefit from your changes. Access to the source code is a precondition for this.
Notably it is NOT the freedom to distribute anything under your name which you did not create; that also includes that changes to content you published under an open-source license (as allowed by freedom 4) must be marked or such versions must at least be clearly marked as modified (and possibly renamed) - yet attribution to your work must be granted. As such "forging" or mis-representation is not allowed with any open-source license.
Also notably, an open-source license intention is NOT to "create trust" with whatever user. It's about enabling people to make their own decision, to apply their own judgement and to not require them to trust anyone.
Now, if you want to require that your work is NOT distributed at all or used whereever without your consent, nor want to allow people to make modification to the sources of your programme, we are not talking of open-source anymore at all - far from it. That's what every proprietary license does. Consult a copyright layer to draft a proprietary license or EULA for you.
As to the "trust issue": you can, of course, make your source code available and impose whatever restrictions you want on its use by your non-open-source license. But unless you allow people to actually build the software from scratch, the "trust issue" is not really resolved either as there is no means for them to verify that you actually provide the source code which corresponds to the binaries you distribute. Debian solves this issue with the reproducable builds.