Evans, a cost savings project aims to reduce all activities, materials and personel that does not, directly or inderectly, add value to the final client. Quality and safety are usually major added value to the final client and reduce its levels only reduce the value that is sold in the end.
Change in culture demands a change in the people's behaviour what demands a proper communication to trigger and sustain these changes. In order to do this task, I recommed you to whatch Simon Sinek TEDx video "How great leaders inspire action" and read his book "Start with Why".

I've been in many organizations & situations where Sr. Management interpreted this differently.
The worst imposed unreasonable restrictions on employee decision making, tightened financial controls on things like expense accounts, or cut headcount & programs that improved employee morale, all in the name of cost savings. see https://en.wikipedia.org/wiki/Albert_J._Dunlap
The best looked at the long term and used programs like 6 Sigma to focus attention on sustainable cost savings projects and encouraged the best ideas to float to the top regardless of the source of that idea. Lower levels in the hierarchy get recognition and technical support to fully explore rough concepts and implement them.
Luis has and excellent definition of cost saving's positive objectives. Improving employee and overall productivity, reducing raw materials used and improving yield are the usual methods to get there. Having competent process experts to explore the best ideas and determine the potential ROI is what some organizations lack in today's environment.

Cost savings projects should be included in a product life cycle plan using tools such as but not limited to lean 6 Sigma or Toyota Production System. Being part of the total plan allows the cost savings culture to develop organically within the organization and not like a forced last minute reactive response.
Unfortunately, what sometimes happens is that cost saving measures are used as a band aid when the original optimistic business case isn't happening as planned.

This is an incredibly broad question. One ca only anser it in some kind of context specifically otherwise folks are ansering but without realizing it it's only from their unique vantage point(s).
In general, all else being equal or imporved post cost savings regarding products and their related services and experiences, one simply cuts expenses to improve absolute profitability, period.
If anything declines per the above, the cost savings are likelly offset both literally and strategically.
Literally by example means if in the moment the product declines in its performance, its user experience and or service. That means fewer folks may buy the product and it may look inferior relative to it's alternatives, be it direct or indirect competition.
Strategically means a long term effect from the former literal, momentary changes. Reputation may suffer, cun sumer allegiance and attitudes may suffer and so on.
The ideal cost saving solutions are where absolute profitability improves as well as both the literal, momemtary changes as well as the strategic.
By example, a firm I founded and that our investment assembly recently sold to Nippon Shokubai regards chemicals that can polymerize and among other things form coatings that cure versus the current methods in seconds to minutes versus minutes and hours, requires no energy input versus the competitors light or usually heat in mile long "ovens" and that require no solvents whereas the alternatuves typically require solvents. These materials cost less than the current products to maake and purchase, deliver superior coatings to anything currently available by example for cars and trucks and overall cut the cost to build and operate an automotive plant in half.
A more interesting example are where a new materials, say an adhesive, costs $100 a pound that replaces one that costs only $1 a pound. In this case, we delivered the same or better overall product but the system manufacturing savings and efficiencies exceeded $250 per pound of purchased $100 adhesive. Witha net saving per pound of $150, the $1 guy has to give away his product and hand over $150 in cash just to be considered.
The key takeaway - costs saving beyond getting a better price on existing purchases should never start in purchasing - it's a persistent, holistic innovation opportunity both internally and even as well to the consumer, be it a firm or an indvidual.