This is a great question, and requires a big in-depth answer so let's break it down.
HOW DID THE INCIDENT IMPACT SHIP DESIGN
The best place to start with this is Titanic's two sister ships, Olympic and Britannic.
Olympic: Was at sea the night Titanic sank, 500 miles away when she received the distress call. After rushing to the scene, she was informed en route that Titanic had sank. Captain Haddock offered to take on survivors, which Bruce Ismay (chairman of the White Star Line) sharply declined due to worry seeing Titanic's almost identical twin would be horrifying to the survivors. Olympic continued on her trip to Southampton. Upon arrival, she was quickly stacked with forty collapsible lifeboats- enough for capacity. Unfortunately, this wasn't enough for nearly 300 fireman, who went on strike and refused to sail on Olympic as she prepared to head to New York about two weeks after the Titanic disaster.
They were right. These collapsible boats were pulled from Navy troopships, and while White Star line had them certified by the Board of Trade, some of them were so old and unused they were collapsing and unable to be unopened. The striking firemen refused to board Olympic unless she was refitted with wooden lifeboats. In response, White Star fired the union men, and hired non union firemen for the NY sailing.
The union intervened, demanding a test of four collapsible boats which they received- finding one to be unseaworthy. Reaching an agreement with the WSL, the union was prepared to allow its men to sail- but now they objected to sailing with non-union crew. Over 50 men left Olympic, stating it was unsafe to sail with non-unionized crew. The trip to New York was cancelled. The 50 were arrested for mutiny, but WSL knew a losing case when they saw one as the public was firmly on the side of the strikers. They were found guilty of mutiny, but received no punishment. WSL- still recovering from the public backlash of Titanic and worried about the optics of jailing safety standard demanding crew- re hired them so Olympic could sail in mid-May.
Olympic sailed over the summer as WSL began the process of designing her refit. In the fall, Olympic returned to Harland and Wolff in Belfast to be redesigned.
First was lifeboats- Olympic was fitted with new Wellin davits that allowed her to carry sixty eight lifeboats, forty-eight more than the twenty she was carrying before the Titanic disaster.
The next were her watertight bulkheads. A large contributor to Titanic's demise was how easy it was for water to pour from compartment to compartment as her bulkheads only extended to E-Deck. Olympic's were bought up to B-Deck, and an extra was added- giving her seventeen watertight compartments and allowing six of them to be flooded while she still remained afloat, including hold #1 on the orlop deck being completely watertight. Lastly, all of this was encompassed in a double hull (varying between 2 feet and 4 feet), a major upgrade from the initial design which had the sisters with a double bottom only.
Of course, WSL's desperation to win back public confidence (and stave off the financial bleeding that resulted from Titanic) meant that Olympic's redesign came at a cost of passenger experience. While they did take the opportunity to re design some of Olympic's interior, having bulkheads extend all the way to B-Deck suddenly made it trickier for passengers to navigate what was already a labyrinth of a ship.
You can see the the difference between 1911 Olympic and 1913 Olympic. I'll also provide this diagram showing her re designed watertight compartments.
Incredulously- the NYT ran an article in the spring of 1913 (which I'll provide at the end of this response) touting Olympic's refi, saying that "Harland & Wolff's naval architects believe that they have realized the quest of an unsinkable ship."
Britannic, meanwhile, had been laid down in November of 1911 and by the following April, only her keel and her framing was being worked on so it was much easier to re design her. Along with all of the new features added on Olympic, her beam was extended, she added a new turbine giving her more horsepower than her sisters. Most strikingly was the addition of crane davits, allowing the storing and lowering of 6 lifeboats each and, in a safety feature not featured on either of her sisters, these davits were electronic. She got five of these installed before she was recruited into WW1, and so was completed with the original, hand cranked, Wellin davits.
What was even more remarkable about these davits is that they were large enough to reach lifeboats on the opposite side. This meant that, in the event of disaster and a severe list (which her doomed sister avoided), these davits would be able to reach across for loading. Good thinking too, as Britannic's eventual sinking did have her sinking with a severe starboard list.
All of these upgrades had Britannic sailing with over 1000 more spaces than passengers she could carry. However, her eventual fate as a hospital ship meant she ended up carrying more people than she was designed to as a passenger liner.
Lastly, there is enough evidence that Britannic's name was originally supposed to be Gigantic. Certainly, it fit the theme- The Olympians, the Titans, and the Giants much better than... The Brittons. However, she was being referred to as Britannic before the Titanic disaster, so the story of her name being changed due to the sinking is probably not accurate.
In an article published in the New York Sun in May of 1912, Bruce Ismay denied that the new liner was ever supposed to be named Gigantic. Unfortunately, documentation exists that it did. I would venture to guess that Ismay, dealing with Titanics fallout, was eager to quash any inkling that WSL was too big for its britches, and to focus attention on safety... not size.
HOW DID THE INCIDENT IMPACT RULES AND REGULATIONS FOR OCEAN LINERS
The changes to Titanic's sisters were absolutely an attempt by WSL to gain the public's trust back, especially as customers ran to competing shipping companies- particularly rival Cunard. However, by 1914, all passenger liners were now subject to strict new rules of operation under the International Convention for Safety of Life at Sea or, SOLAS.
The SOLAS treaty, recommended and undertaken after the American and British Inquiries into Titanic, regulates the safety standards for passenger ships. SOLAS' first act was to guarantee lifeboat space for every passenger aboard, as well as 24 hour wireless radio service, with backup battery and licensing for operators, in conjunction with the Radio Act of 1912 (due to the Californian incident with Titanic). Also on the docket, and due to Californian was the establishment of red rockets being the signal for distress.
Immediately after Titanic, the navy began patrolling the upper east coast of North America for icebergs, which eventually lead to the International Ice Patrol. Unfortunately, neither the Navy for the Coast Guard were able to spare the resources for constant North Atlantic surveillance, so along with the above safety standards, SOLAS created the Ice Patrol with a fleet dedicated solely to iceberg tracking and the protection of passenger ships.
Lastly, while the connection may be tenuous, in 1920 Congress passed the Merchant Marine Act (or "The Jones Act"). While predominantly focused on regulating commerce, it did include a bill protecting seamans rights, allowing crew members to sue for damages due to negligence of the Captain and or shipping company. While I can't say that Titanic was definitely connected to the passing of the Jones Act, WSL's absolutely brutal litigation tactics post sinking no doubt fueled a fire that had been building.
SOLAS is still in effect today, protecting safety standards in shipping. When you board a ship today, you will have a guaranteed seat on a lifeboat and an immediate safety drill- all due to the Titanic disaster. The International Ice Patrol still monitors the North Atlantic both by air and sea and works out of New England and Newfoundland.
Further reading and my sources for this post are two wonderful books by Mark Chirnside, RMS Olympic: Titanic's sister and The Olympic Class Ships: Olympic, Titanic, and Britannic. Also referenced was CALL OLYMPIC UNSINKABLE- New York Times, March 13th, 1913, and GIGANTIC TOO BOMBASTIC- The New York Sun, May 16 1912. You can also read the Radio Act of 1912 here
How did the changes to the rules for ocean liners affect other vessels? Freighters, for instance? Coastal vessels? Lakers and riverboats?
I ask because I had always heard that the laker Eastland sank BECAUSE of the extra lifeboats added after the Titanic. She became unstable and unsafe, killed more people than the Titanic after she capsized in Lake Michigan.