Genetic selection
Genetic selection (even manipulated by selecting "the best" seeds for a particular trait) is not excluded from organic growing - this is a technique that has been used by farmers for many thousands of years and just on a personal level I use it in my home garden to grow heirloom vegetables (open pollinated, old varieties) to select plants that grow best in my area ... they are still the same vegetable, but my 5th generation of that vegetable is going to be slightly different (and more valuable to my neighbours) than the same vegetable grown in the same way a thousand kilometres from me (this old technique is called localization, and is a form of genetic selection, rather than direct modification).
Grafting
I also use grafting, to join the two plants from the same plant family (usually from the same species, but not always). An example of how this is used is to control the rate of growth of a fruit tree - the dwarf fruit trees that you can buy are grown on a very sturdy rootstock (the part that grows in the ground), and the grafted top is from a tree that might otherwise grow too tall to easily harvest, but has the very best fruit. This technique is used for a variety of reasons, including to provide disease resistance or to allow a plant to grow in conditions more difficult than the standard tree could tolerate. It can also be used for vegetables.
The important point with this is that the genetics of the fruiting top remain the unchanged by growing it on a different rootstock (which is how multi-graft trees work - you can have many different varieties of fruit on the one tree - they keep their genetic differences, so the fruit of each variety remains distinct from one another). Grafting is also allow for organic crop growing.
Hybrid
Hybid vegetables are fairly common, and it is the original way that our (now) heirloom varieties of fruits and vegetables came about. A hybrid is where two known varieties are grown next to each other and cross-pollinate. The seeds of this cross pollination is a hybrid, and when grown on will be like children - they will each have varying traits they've inherited from their parent plants. For crop growing, some of these crosses are valuable (they may be juicier, or stronger, or bigger, or tastier ... etc.), but the seeds of these first crosses tends to be variable (it takes a few generations of growing to stabilize traits), so there is a hybrid numbering system used to denote that the plant is a hybrid, which tells you that if you save the seed and grow it on, it may or may not (most likely not) come out like the original plant. On a commercial level, the cross-pollination process is industrialized (but effectively the same concept) - this is still plant breeding and is acceptable for organic growing.
Genetic modification
Genetic modification is where gene's within the organism are directly manipulated (rather than using breeding techniques)
In the example you gave, roundup ready crops were genetically modified to gave glycosphate resistance using genetic traits of a bacteria.
The concern is that these modifications are transferable and inheritable (for plants, wind and insects are two common vectors).
Glyphosate
Glyphosate itself has proved to be more toxic to mammals than originally thought (the pesticide has been in use for about 40 years).
Aside from any cancer concerns for users of the product, the product doesn't discriminate by insect species, so all insects (including bees) are destroyed.
In addition to this, there is an increased risk of pesticide residue on crops that have been genetically manipulated to resist the pesticide. A withholding period (which is the amount of time prior to harvest that a crop is not to be sprayed) is used for food crops, but there have still been instances where glyphosate residue has been found in products made from these crops.
C
RISPR/cas9 & cas13
This is an even newer form of genetic modification, that allows for very rapid (and much cheaper) development of genetic modifications.
Genetic modification is not restricted to plants. The ethics around gene technologies are still being debated, and vary between countries.
https://www.gizmodo.com.au/2018/12/...-losing-track-of-gene-edited-crispr-patients/
Where the problem currently lies with GM, is that we simply don't know enough about the permanent outcomes of the changes we are making (yet).
CRISPR, for gene splicing, has been around for less than a decade.
This might be something you'd like to read:
https://www.vox.com/2018/7/23/17594864/crispr-cas9-gene-editing