If they don't want the signed contract back, that's the first time I've seen it. Every employer I worked for wanted the contract back with a signature. Newer ones were OK with an electronic (not digital, just pixels instead of ink) signature but they all wanted a signature.

When they get the modified contract back that's their notification of your adjusted terms. Is it polite to ask them to print a modified contract? Yes. But it may not always be practical.

[This is not legal advice, duh]

Because employer/ employee is asymmetric (unless you're literally hired by an individual) my understanding is that it is good sense for them to prefer your amended contract if the terms you wanted are acceptable, because the law in many places says if you have a take-it-or-leave-it contract then the person in the "take it or leave it" position, here the employee, is entitled to interpret any ambiguous provisions in the most advantageous way. Having amended the contract, you are now on equal footing with the employer and any remaining ambiguity is resolved equitably which means less risk of nasty surprises for them.