"As many as are led by the Spirit of God, they are the sons--and daughters--of God. For ye have not received the spirit of bondage again to fear; but ye have received the Spirit of adoption, whereby we cry, Abba, Father. The Spirit [Him]self beareth witness with our spirit, that we are the children of God:" (Romans 8:14) Impel
"For it is God which worketh in you both to will and to do of His good pleasure." (Philippians 2:13)
The implication here is that whatever it is we feel like doing for God is coming from within. The prefix "im-" being essentially the same as "in-" denoting we're being led of "the Holy Ghost which is given to us." (Romans 5:5) It really is the best way when you think about it. When you've reached a point in your life where you're--not not submitted to someone else--but free to obey the leading of His Spirit in "whatever thou settest thine hand to." (Deuteronomy 28:8) It carries with it all the responsibility you've been accruing throughout your life. If you're not sure you're there but you want to be, let fly. Tell God you want to be submitted to Him only. And He'll be sure and bring you in and among those from whom you're destined to take what you need and then be on your way. We all complement one another. And compliment, too.
"I will instruct thee and teach thee in the way which thou shalt go: I will guide thee with mine eye."
Compel
"Be ye not as the horse, or as the mule, which have no understanding: whose mouth must be held in with bit and bridle, lest they come near unto thee." (Psalm 32:8-9)
Your dictionary pretty much synonimizes the two words. Compel, however, at least to my mind, connotes a more outward exertion toward that which you're supposed to be doing. And it should go without saying that as God's kids, we would seek to be doing His will. Getting more and more dialed in as the days go on. Sometimes though, God has to come across as the very thing we're afraid of in order to show that it's us who has the misconception. Does this make sense? I find that the very people who show a predilection or tendency to act in the ways that I-do-but-am-blind-to, are the very people that God chooses to speak through and bless me. In spite of my bias to only want to hear from a "me approved" source. And so how do I pay them back? ("Owe no man any thing, but to love one another..." Romans 13:8) By praying for them of course. And God will do the very thing for me that I choose to exert my breath and time and life through prayer--for someone else.
"Therefore I take pleasure in infirmities, in reproaches, in necessities, in persecutions, in distresses for Christ's sake: for when I am weak, then am I strong. I am become a fool in glorying; ye have compelled me..." (2 Corinthians 12:10-11a)
Propel
"Go ye into all the world, and preach the gospel to every creature." (Mark 16:15)
"Preach the Gospel at all times and when necessary, use words." St. Francis
The two go hand in hand. When Jesus says "Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of Heaven; but he (and she) that doeth the will of my Father which is in Heaven." (Matthew 7:21), He's levelling a hard statement into the reader. But the thing is, is it hard because we say it is? Or is it stark and dry because we're used to the other way around? If His statement of making Heaven only by walking in the Will of God sounds harsh, perhaps I should reorient my understanding and opinion of "harsh". My view of what is "hard" and what is "easy" is colored by my understanding of just how gentle and sweet "Our Father who art in Heaven" is. He's sweeter and gentler than anyone has ever represented after Christ walked the earth. The key to knowing that you are the will of God, is by taking everything you can to Him for scrutiny and getting His opinion. Don't worry. He's not going anywhere. Even as He sends you out to those you're called to affect, He is always the same, and always in the same place. He is the stillness in the world.
"Therefore I esteem all Thy precepts concerning all things to be right; and I hate every false way." (Psalm 119:128)