Answer
You've raised several different points:
1. The Holy Spirit indwells believers and God is omnipresent. The Holy Spirit did not indwell old covenant believers.
Yes, the Holy Spirit indwells believers, and God is omnipresent. But I disagree with the Old Testament scholars who deny that the Holy Spirit indwelt Old Testament saints. Since one must be "regenerate" or "born again" in order to have faith (the doctrine of total depravity prohibits any other view even in the Old Testament), Old Testament saints must have been regenerate in order to believe. In regeneration, our spirits are made alive only because they draw life from the Holy Spirit who indwells us (Rom. 8:9-11). Thus, Old Testament saints must have been indwelt by the Holy Spirit in order to believe.
In the case of David, when he spoke of the possibility of God taking away his Holy Spirit, he was not speaking of the indwelling presence of God. Rather, he was speaking of God's anointing him as king. David did not want God to take the throne away from him. As you might guess from this interpretation, we teach that the Davidic covenant was conditional (e.g. 2 Chron. 6:16; Ps. 132:11-12), as were all Old Testament covenants. Meredith Kline's view (which set the standard in Reformed circles many years ago) differs from this, but the Ancient Near-Eastern evidence of the existence of unconditional treaties/covenants (i.e. royal land grant treaties) has since been debunked. Notice that all Old Testament covenants contain stipulations and curses, which logically would not belong in "unconditional" covenants.
2. God is manifest in a special way when believers worship together.
I believe this is true, but I'm not sure Matthew 18:20 is the best proof text for this. That text falls in the context of church discipline, and seems to emphasize authority more than actual presence. In other words, Jesus backs up their determination. I would prefer to prove this from passages like 1 Corinthians 3:16-17. There, Paul speaks of the church (i.e. a body of believers, not a building) as the temple of God, indwelt by the Holy Spirit. The significance of this passage is that it equates the church with the Old Testament temple where God's presence was manifested and where his name dwelt. Although we no longer see the shekinah glory, which was a mere forshadowing of the glory to come, we have something even better -- we share in Jesus Christ himself (1 Cor. 10:16).
3. Should believers expect and anticipate God's manifest presence every Lord's Day?
Yes, assuming that by "manifest" you don't mean "visible." I would also add that this presence may not necessarily be sensed by everyone present. We often don't sense God's presence indwelling us at every moment of our lives, and we often don't sense his omnipresence. God need not be sensed by us in order to be really present in worship in a different way than he is present at other times.
Part of your question might be inferred to be asking if God can be more or less present in any particular place. The Bible does distinguish between his dwelling in us (Rom. 8:9-11) and his omnipresence. If such a distinction can be made, then it seems to me that his special presence in believers differs from his omnipresence in the world. Since this different presence is possible, there is nothing to preclude the possibility of other forms of his presence. Another way to see this in the Bible is to note the difference between God's unseen presence with Israel, his visible presence in the glory cloud, his more dangerous and more glorious presence in the Holy of Holies, and finally his even more glorious revelation to Moses in Exodus 33. Behind all these also lay God's omnipresence, and his Spirit indwelt the various believers such as Moses.
The Bible seems to me to indicate that varieties of God's presence also exist in the New Testament. Again, I would go to 1 Corinthians 3:16-17 to draw the Old Testament parallel, and also to passages which speak of the spiritual gifts. Since spiritual gifts demonstrate the actions of the Holy Spirit, and since such gifts include the preaching of the word and some of the other things manifested in a church service, I would suggest that these regular events indicate the special presence of God in our services. Finally, we generally pray to invoke God's presence in our services, and I believe this is a reasonable prayer which he is inclined to honor.
Answer by Ra McLaughlin
Ra McLaughlin is Vice President of Finance and Administration at Third Millennium Ministries.