Yes, entering the spell's area only applies the effect "when a creature moves" there by itself
Your DM is right that "when a creature moves" exclusively refers to the creature moving itself using its "Movement, Action, or Reaction" which all rely on its Speed. That is, it doesn't include if a creature is hurled or pushed into the area (though it can be a nice setup for the spell effect to apply when the creature starts its turn there).
"When a creature moves" is what would also provoke Opportunity Attacks
The rules on Opportunity Attacks (PHB p. 195) defines what "when a creature moves" does and doesn't entail in D&D, emphasis mine:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. (...)
You can avoid provoking an opportunity attack by taking the Disengage action. You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction. For example, you don't provoke an opportunity attack if an explosion hurls you out of a foe's reach or if gravity causes you to fall past an enemy.
That is, "when a creature moves" is defined such that:
- It involves that the creature either uses its "Movement" (only on the creature's turn), "Action" (e.g. the Dash action), or "Reaction" (e.g. the Ready action, Dissonant Whispers, a Battle Master's Maneuvering Attack, a Scout's Skirmisher feature, or a Glamour Bard's Mantle of Inspiration). As a sidenote, the creature doesn't have to move willingly in the sense that the effect still applies if it uses its "Movement, Action, or Reaction" to move due to a Dominate spell, Confusion, or similar effects.
- It doesn't include when "when someone or something moves you", such as "if an explosion hurls you".
Movement is a game term defined under Movement and Position (PHB p. 190):
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.
Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.
Thus, Movement happens on your turn and uses your Speed in the form of walking, jumping, climbing, or swimming. Likewise, all of the examples I can find for using your Action or Reaction to move are also limited by your Speed (teleporting is explicitly excluded from the definition in the Opportunity Attacks section).
The section on Opportunity Attacks is the only relevant guidance from WoTC on what "when a creature moves" does and doesn't entail. Thus, the same guidelines should apply to Sickening Radiance and other instances of said wording. (Keep in mind that Disengage doesn't help you against Sickening Radiance but is specific to avoid provoking opportunity attacks.)
"Enters" is used instead if application "is more open-ended"
Are the terms "moves" and "enters" interchangeable or deliberate wording? Though only considered advice and no longer official rulings, in a tweet (thanks J. A. Streich), Jeremy Crawford clarifies that the distinction between "moves into" and "enters" is used deliberately as the two terms have different meanings, emphasis mine:
"Enters" and "moves into" mean enters and moves into, respectively, in D&D. They have no special game meaning, other than that "moves" refers to movement. "Enter" is more open-ended.
"Moves" refers to Movement which is a game term relying on a creature's speed as I account for above. On the contrary, a spell description will instead use the more general term "enters" if the spell effect is also supposed to take place when a creature is hurled or pushed into the spell's area as explained in Sage Advice 2016 regarding Moonbeam, emphasis mine:
Our design intent for such spells is this: a creature enters the area of effect when the creature passes into it. Creating the area of effect on the creature or moving it onto the creature doesn’t count. If the creature is still in the area at the start of its turn, it is subjected to the area’s effect.
Entering such an area of effect needn’t be voluntary, unless a spell says otherwise. You can, therefore, hurl a creature into the area with a spell like thunderwave. We consider that clever play, not an imbalance, so hurl away!
Sage Advice carefully outlines that this timing applies to spells such as "blade barrier, cloudkill, cloud of daggers, Evard’s black tentacles, forbiddance, moonbeam, sleet storm, and spirit guardians" - which is the complete list of spells in PHB for which the spell description uses the term "when a creature enters the area for the first time on a turn or starts its turn there". This consistency underlines that the wording is always used deliberately.
Furthermore, if the spell effects of Sickening Radiance and other spells that instead use the term "when a creature moves into the area for the first time on a turn or starts its turn there" where likewise supposed to apply when a creature is hurled into the area, they would have been included in said Sage Advice list of spells "with the same timing as moonbeam". Thus, by carefully omitting any spells that use the wording "moves into" rather than "enters", WoTC implicitly tell us that the two terms are not interchangeable but those spells should be handled differently. Apparently, WoTC thinks the section on Opportunity Attacks suffices as the only official guidelines on how to interpret the term "when a creature moves".
(The mechanic that "when a creature moves" triggers a spell effect is exclusively used for the following list of spells in PHB, XGE, and TCE: Booming Blade, Create Bonfire, Healing Spirit, Investiture of Flame, Sickening Radiance, Snare, Spike Growth, and Transmute Rock).