I don't like tshirts whose logos melt when you iron them, or crack if you put them in the dryer for too long.
What do you call the printing process where the logo becomes part of the shirt itself, and is less susceptible to damage?
I believe what you are referring to is heat transfers from off the shelf that use an iron on graphic which is designed for personal usage.
There is also sublimation printing which still uses heat to transfer and typically the graphic is digitally printed on a special film:
DTG (direct to garment) uses spray on ink and typically a heat press to assure dried ink.
All printing techniques are subjective to damage but if you are looking for the best print quality and life screen printing is typically suggested over them all. Screen printing doesn't fade as much as DTG printing but there are pros and cons for them. If you have a full color or technically anything over 3 colors some professional shops can create accurate color results for shirts with screen printing but it is hard. Also, to add to that some shops require a min. to screen print because the setup is intuitive and not usually worth doing a run of less than 25.
I believe these were originally found on Pinterest but per someones comment they were created by Row Apparel
Screen printing definitely sounds like what you're looking for. In addition to what's already been said, another aspect you should take into consideration is the type of ink used for the prints.
For t-shirts, the two most popular types of inks are water-based and plastisol inks. Water-based will typically have a better penetration of the fabric, which will help it feel as if it's "part of" the shirt. If you run your fingers over the print, it should feel soft. Plastisol has less penetration and tends to sit on top of the fabric and will give the print a different texture than the rest of the sure. It will have a "plastic-like" texture to it and have a heavier feel.
Since plastisol is usually easier to work with for the printers, it's often cheaper. Before having prints done, verify with your printer what type of ink is being used if the texture is important to you. It also doesn't hurt to ask for a sample print so you know exactly what you're getting. Don't expect a sample with your artwork; an overrun from a similar job would be fine so you can gauge the quality.