As always, compassion is the important thing
When explaining something you see as a 'fact' to people who disagree as to whether or not that fact is the case, be prepared for blowback. People don't like it when 'facts' are questioned because it disrupts their surety that their understanding of the world is accurate overall, which reminds them that they are in despair 1.
People also don't like being told that they are wrong about something, especially when they believe that the wrongness is obvious or that others see the wrongness as obvious. We put a lot of value on rightness in thinking, and many people expect to be attacked if they are caught out as having been 'stupid'. People are very afraid to be labeled 'stupid'.
All of this defensiveness is exaggerated on the internet, with its lack of faces to read or tone to hear, and with the anonymity it gives people who wish to act uncharitably. People, as much as they hate being made wrong and on the basis of that wrongness being devalued, are quite happy to make others wrong and on the basis of that wrongness devalue them, if they think there can be no retribution.
Therefore, when correcting someone, it is important to assure them this is not the case, that you are not attacking them and wish them no harm. However, you must be subtle. Do not say things like "I don't think you are stupid" or "Take heart! Your value as a person far outweighs any value I assign you on the basis of intellect: even were I to discredit you the latter, I would nonetheless acknowledge the former". People, especially on the internet, will not receive these as statements in earnest.
Instead, be gentle in your phrasing and terms. Put uncertainty into your claims even where there is none, to remove the appearance of power over the person to whom you speak. Take the time to connect with the person and express approval for their reaction to whatever criticism you do give, if appropriate. Ooze as much compassionate, caring tone you possibly can without coming across as 'fake' or 'condescending'. Once you've convinced someone that you are safe and then convinced them to stop being defensive, they can begin listening and changing their opinion. At that point, you don't need citations or aggressive rhetorical arguments or a hundred like-minded persons backing you up, because the other person isn't fighting you anymore.
This is the best way, in terms of helping the person you are seeking to correct, but it is also very emotionally and temporally expensive. It's not always the right tool for the job, especially if it's likely (like in this case as of 2017) that the position is wholly out of ignorance and has, as of yet, no value to the correct-ee whatsoever.
An efficient, low-cost implementation of this option is to treat the exchange similarly to a rules conflict in a game:
whenever I encounter a situation where someone has made an important declaration that I think might be in error, I say something like:
"I think I'm confused." - I always assume that I might be wrong (even when I'm pretty sure I'm not.) I would then follow up with something like "I thought that it was vs. Dex, not Str. What am I missing?"
Then, I gracefully accept whatever answer I am given...
You might not always convince everyone, but ultimately I don't think conversion rate is the best value criterion to use for this sort of discourse.
A formulation to address movement specifically:
"I think I'm confused. I know 3.5 had move actions, but I didn't think 5e did. What am I missing?"
You should trust the person you are talking to do the research from there. You don't need to cite things at them because they are capable of finding the answer themselves without too much work, and they can ask for a citation if they want one. Citations detract from the sincerity of your feigned confusion.
If you absolutely must, you could do the following:
"I think I'm confused. When I read about actions on pg. 190ish of the PHB, I didn't see anything about move actions. I know they are a thing in D&D 3.5, but are they in 5e as well?"
It is desirable to avoid the following:
"I think I'm confused. Page 189 of the Player's Handbook states in part:
On your turn, you can move a distance up to your speed and take one action.
Since movement and action taking are explicitly called out as separate, it would seem they are not the same, which would seem to invalidate your answer, which states (quoted part of the answer which indicates movement is an action or refers to a 'move action' in a problematic way), which seems to indicate that they are the same. Since it can't be both an action or not an action, you or the PHB must be wrong. What am I missing?
This comes across as aggressive, despite the bookends. You don't need to support your claim so thoroughly when your claim can be evidentiated easily by the other party. It has a certain "let me give you a thorough, itemized list of everything wrong with your post" feel. That's not helpful.