"Propaganda" is in the eye of the beholder. There is the translation issue, so I don't know for sure how much of this applies to the Russian word he is using, but "propaganda" has the same root as "propagate". Its literal meaning is any speech or act that causes a view, attitude, cause, group, activity, or belief to be propagate or be viewed more favorably.
So anything that causes gay people to be viewed more favorably is "propaganda". "Gay people are not disgusting perverts" is propaganda. "Gay people shouldn't be killed" is propaganda. Putin is claiming (or, at least, his words are being translated as saying) that he doesn't have a problem with gay people, but he does have a problem with people saying they shouldn't be killed, which is absurd.
Now, of course, the connotations of "propaganda" are more specific: the connotation is that it's deceptive speech meant to manipulate people into accepting something bad. This gulf between the connotation and denotation of "propaganda" allows people to engage in equivocation. When it comes to defending their position, they rely on the connotation of "propaganda". But when it comes to actually implementing the policy, "propaganda" is so broad as to allow almost any conduct to be included.
This does have some similarity to DADT, but there are several differences. First, DADT was just with the military. People who didn't want to live under DADT could choose to not enlist, while the anti-gay laws of Russia apply to everyone in Russia. Second, the consequences under DADT were just discharge from the armed services, while Russia imposes criminal penalties. Third, DADT was a move towards loosening the restrictions. Clinton wanted to get rid of the restrictions entirely, but didn't have the pull to do so entirely, so he reached a compromise that allowed him to seriously weaken them. Putin, on the other hand, is increasing the restrictions on gay people, and it's clear that he is doing so either with malice towards gay people, or to pander to people with malice.