First off, the teaser did not lie, there was no editing trick, Goh does indeed catch the Suicune while the poachers had it trapped.
However, while Goh ran to safety as Ash fended off the poachers, he trips and Suicune lets itself out of the Poke Ball where it then runs away up a tree. Goh follows it and sees it's being defended by a batch of Pokemon in the tree. As Ash gives the poachers the slip, Goh and his Pokemon attempt to show them mean no harm, first by offering Suicune Pecha Berries to cure any Poison it has, then HP healing berries to both Suicune and the Pokemon guarding it, finally Goh notices it's leg is injured and after some hesitation from Suicune allows him to use a Potion on it. The poachers eventually find them and Goh tries defending Suicune but gets overwhelmed. Ash comes to help but Suicune gets its second wind and freezes the poachers who eventually are hauled away by Officer Jenny.
And with the poachers gone, Goh releases Suicune.
ONLY for Suicune to then decide it'll stay with Goh! So yes, Goh has indeed caught Suicune! And though it trusts Goh, I have a feeling if Goh wants to use it to battle he'll probably first need to build more trust with it. Though it'll likely just stay at the Lab most of the time.
BONUS: With that all said and done, this episode does also answer a major question many people are thought has been a "plot hole" over the years: Why don't the villain groups just use a Poke Ball to catch the Legendaries? For a while the fanon excuse was the Poke Ball acts as a power limiter and the villain team wants to use the Legendary at its full power. However, the anime has given a much more logical answer: just because someone catches them doesn't mean they'll listen. Plus with the ability to pop out of the Poke Ball anytime they please, the villain team catching the Legendary in a Poke Ball will do diddly-squat as it still won't listen to them and can still escape. Better to focus their energy on a method that will at least bind the Legendary. So good job writers on settling that debate.