I've said it before and I'll say it again here, I don't know much about this method. I get the chemistry behind using just conditioners to cleanse one's hair, but I don't get the rules and regulations that seem to come with the various theories or methods. There are tons of ingredients that can cause build up, but it seems the ones to avoid are oils and silicones, or cationic polymers or cationic guar or other conditioning agents. Or maybe not. I simply don't know. For every person who asks me to justify suggesting the use of silicones for this method, someone else writes to me to tell me they love silicones. For every person who avoids oils, others swear by them. So it's really about the philosophy to which you plan to adhere, and learning more about that specific program.
So how does a cleansing conditioner differ from a normal conditioner? It doesn't. For the most part, the cleansing conditioner contains the same ingredients as a normal conditioner, although I've seen a few with very low levels of foaming surfactants like cocamidopropyl betaine. I have noticed that most of the conditioners with the word "cleansing" in the title tend to stay away from silicones (although the Wen conditioner I mention below doesn't), but that might be because those I've seen also bill themselves as natural or organic. I've also noticed that a lot of them contain mint, perhaps for the tingly feeling it can leave behind? (Click here for the longer post on duplicating cleansing conditioners.)
If you can't have silicones, then get a conditioner without silicones or make your own. If you can't have oils, then avoid anything that contains oils, butters, fatty alcohols, fatty acids, and so on (or perhaps it's just normal oils - this is one area that really confuses me). You're trying an approach that requires you to do a lot of work learning and experimenting and generally spending a lot of time and energy on figuring out what your hair likes and doesn't like, and the only way you can learn more about it is to read a lot and experiment a lot. Keep really accurate notes about the various ingredients and what impact they've had on your hair. If you want to avoid something like silicones or cationic polymers, then you'll have to spend the time learning what they are, how they show up in our conditioners, and which ones can create build up (and if they do, how to remove it). I'm afraid I can't offer a one sentence answer or a rhyming couplet to remember what you can and can't have as every person is different.
If you take a look at the post in which you made the comment, I take a look at two conditioners. One calls itself a cleansing conditioner and the other doesn't. One is expensive and one is reasonably priced. Both contain silicones. I don't actually recommend either of them - I don't recommend commercial products, I recommend making your own! - but use them as tools to explain why both would be suitable as a co-washing conditioner.
If you are interested in this method, want to know more about how shampoo works on our hair, or just feel like reading more today, here are a few suggested posts...
How does shampoo clean our hair?
How does the no shampoo concept work?
Is shampooing necessary for our hair?
What is build up?
Duplicating the Curl Junkie Daily Fix Cleansing Conditioner
If you're on the Dish forum, then you know I'm leading a crusade to stop the shortening of shampoo to 'poo for what I think are really obvious reasons. I implore you to write those four little letters to make something we wash our hair with sound icky!