Is that all your dictionary says??
It usually means to begin, but if someone stops or hesitates seemingly without reason, "Proceed!" could be used to encourage him to continue on. "Headsman, proceed with the execution."
When the beginning of some event is in doubt because of something unexpected (rain at a baseball game), someone might well say, "I think we should proceed." This would be appropriate at the very beginning, or following an interruption by a brief downpour.
"Continue" would be appropriate only for something already begun.
"I think we should contunue playing until it's too dark to see." (We have not yet stopped playing.)
"Joe is going to run home and get another ball to replace the one you hit in the pond. Then we can continue playing."