Not exactly.. There are many types of exosomes, however the way you describe it is not how I have learned it to be.

The cell would not send out an exosome with toxin in it to spread the toxin to cause damage only to communicate to the cell as a forewarning mechanism.

That is very similar to the misconception that the virus theory implies, this can be explained and is understood differently.

The exosome that would be forewarning would either be an mRNA or microRNA or other RNA encoded enzymes to tell the cells to respond a certain way and do a certain thing. These exosomes with genetic sequences inside program cells or gives commands to cells and the cells must follow the commands.

These types of exosomes full of genetic sequences made from healthy DNA would not spread the toxin.

However there are ways this process can cause issues.. Say the DNA is damaged that gave the exosomes the instructions to be delivered.. This could end up causing the cell to produce or express a misfolded protein or a harmful chemical that is not actually needed.

Or maybe the genetic information is correct and chemical is needed but the cell is lacking magnesium or other amino acids and so the healthy cell can't make what is needed and so it tries to produce it but it can't so it makes an alternative or lower amount then what is needed.

Or decides it should first make cytokines to go and collect the needed minerals and this looks like a virus because the cells are causing healthy cells to break down in order to collect the needed nutrients from them to do a process in another part of the body. This is one way people develop pain and inflammation symptoms. There are many other ways also.

Basically unhealthy, malnourished and vitamin deficiency can cause errors or problems with human functioning. This is why we eat and exercise and communicate love and laugh but also these things use up what you need to stay healthy.

Humans can rust it's called oxidative stress it's also like having stagnant water or dirty water which creates heavy oligomers which is like if you are repairing your cells with old rusted parts. Clean terrain means the parts are kept light and clean and more flexible etc.

There is a balance we try to keep at some are better at this than others.

Now imagine this complex process happening 24/7 constantly in the body and a body was malnourished or deficient in needed minerals and vitamins then this process can go wrong as the wrong enzymes could be made from the creation of them using amino acids..

This is what virologists call viral infections and is why normally a doctor would tell you to take it easy, drink water and rest… so the body has time to heal the damage without causing more damage.

This is a toxic load or deficiency problem and the body would have created this problem from within however the exosomes can mediate a faulty genetic code sequences.

Now there are exosomes that can clean up cell debris and toxic antigens and signal to make larger junk collectors like phages, bacteria and parasites to collect the rotting cell tissue or other things in the body. This is an extremely healthy process, not one that necessarily should be stopped. A human is getting rid of waste for a reason.

To start understanding the way these exosomes can affect others, have a read through this.
https://www.peace-is-happy.org/post/exosomes-pheromones-and-the-placebo-or-nocebo-effect